The goal of this clinical research study is to learn if carfilzomib and vorinostat combined with gemcitabine, busulfan, and melphalan with a stem cell transplant will help to control multiple myeloma (MM). Researchers also want to learn about the safety and effectiveness of this combination.

Busulfan Test Dose:

Participant will receive a test dose of busulfan by vein over about 60 minutes. This low-level test dose of busulfan is to check how the level of busulfan in participant's blood changes over time. This information will be used to decide the next dose needed to reach the target blood level that matches participant's body size. Participant will most likely receive this as an outpatient during the week before they are admitted to the hospital. If it cannot be given to participant as an outpatient, they will be admitted to the hospital on Day -11 (11 days before participant's stem cells are returned to their body) and the test dose will be given on Day -10.

About 11 samples of blood (about 1 teaspoon each time) will be drawn for pharmacokinetic (PK) testing of busulfan. PK testing measures the amount of study drug in the body at different time points and will help the study doctor decide what participant's dose of busulfan in this study should be. These blood samples will be drawn at various timepoints before participant receives busulfan and over about the next 11 hours. The blood samples will be repeated again on the first day of high-dose busulfan treatment (Day -8). A temporary heparin lock line will be placed in participant's vein to lower the number of needle sticks needed for these draws. If it is not possible for the PK tests to be performed for technical or scheduling reasons, participant will receive the standard fixed dose of busulfan.

If participant receives the busulfan test dose as an outpatient:

On Days -12, -11, and -10, participant will receive palifermin by vein over about 30 seconds each day to help decrease the risk of side effects in the mouth and throat.

Participant will be admitted to the hospital on Day -9.

If participant receives the busulfan test dose as an inpatient:

On Days -14, -13, and -12, participant will receive palifermin by vein over about 30 seconds each day to help decrease the risk of side effects in the mouth and throat.

Participant will be admitted to the hospital on Day -11.

Study Drug Administration (for all participants):

With stem cell transplants, the days before participant receives their stem cells are called minus days. The day participant receives the stem cells is called Day 0. The days after participant receives their stem cells are called plus days.

Beginning on Day -9, participant will swish the liquids caphosol and glutamine in their mouth 4 times a day, for about 2 minutes each time. Participant will swish these liquids every day until they leave the hospital. These drugs are used to help decrease the risk of side effects in the mouth and throat.

On Day -9 through Day -2, participant will receive dexamethasone 2 times a day by vein over about 10 minutes.

On Day -8 through Day -2, participant will take vorinostat by mouth, with food.

On Day -8, participant will receive gemcitabine by vein over about 4 hours.

On Days -8, -7, -6, and -5, participant will receive busulfan by vein over about 3 hours.

On Days -7 and -6, participant will receive carfilzomib by vein over about 2-10 minutes.

On Day -3, participant will receive gemcitabine by vein over about 4 hours and melphalan by vein over 30 minutes.

On Day -2, participant will receive carfilzomib by vein over about 2-10 minutes and melphalan by vein over about 30 minutes.

On Day -1, participant will receive carfilzomib by vein over about 2-10 minutes.

On Day 0, participant will receive their stem cells by vein over about 30-60 minutes.

Participant will receive 3 more doses of palifermin by vein over about 15-30 seconds on Days 0, +1 and +2.

As part of standard care, participant will receive G-CSF (filgrastim) as an injection just under their skin 1 time a day starting on Day +5 until their blood cell levels return to normal.

Study Tests:

About 1 month, 100 days, 6 months, 1 year, and then about every 3-6 months for at least 2 years after the transplant:

• Participant will have a physical exam.
• Blood (about 4 tablespoons) will be drawn for routine tests, to learn how the transplant has taken, and to check the status of the disease.

About 100 days after the transplant, participant will have a bone marrow biopsy and aspiration to check the status of the disease. To collect a bone marrow biopsy and aspirate, an area of the hip or other site is numbed with anesthetic, and a small amount of bone and bone marrow is withdrawn through a large needle. This will be repeated once a year or earlier, if participant's doctor thinks it is needed.

Once a year, participant will have x-rays of all the bones in their body to check the status of the disease.

The study staff will also stay in contact with participant's local doctor to find out if the disease comes back and to check how they are doing.

Length of Treatment:

As part of standard care, participant will remain in the hospital for about 3-4 weeks after the transplant. After participant is released from the hospital, they will continue as an outpatient in the Houston area to be monitored for infections and transplant-related complications.

Participant will be taken off study about 2 years after the transplant. Participant may be taken off study early if the disease gets worse, if intolerable side effects occur, or if they are unable to follow study directions.

If for any reason participant wants to leave the study early, they must talk to the study doctor. It may be life-threatening to leave the study after participant has started to receive the study drugs but before they receive the stem cell transplant because their blood cell counts will be dangerously low.

This is an investigational study. Carfilzomib and melphalan are FDA approved for the treatment of MM. Vorinostat is FDA approved for the treatment of cutaneous lymphoma. Busulfan is FDA approved for the treatment of leukemia. Gemcitabine is FDA approved for the treatment of lymphoma, breast cancer, and lung cancer. The use of these study drugs in combination is investigational. The study doctor can explain how the study drugs are designed to work.

Up to 75 participants will take part in this study. All will be enrolled at MD Anderson.

Inclusion Criteria:

1. Age 18 to 65 years
2. Refractory or relapsed myeloma, defined as one or more of the following: 1. Patients with myeloma treated with first-line therapy including lenalidomide, bortezomib or thalidomide, and one or more of the following: a. Less than partial response to first-line therapy. b. Relapse after 1st line therapy. 2. High-risk disease, defined by del(13q) by conventional cytogenetics, or by del(17p), t(4;14), t(14;16), t(14;20) or 1q+ by FISH. 3. Relapse after a prior autologous stem-cell transplantation (ASCT). 4. Plasma cell leukemia. 5. Plasmablastic lymphoma. 6. Soft tissue plasmacytoma.
3. Adequate renal function, as defined by serum creatinine </=1.8 mg/dL and/or estimated serum creatinine clearance >/=50 ml/min.
4. Adequate hepatic function, as defined by serum glutamate oxaloacetate (SGOT) and/or serum glutamic-pyruvic transaminase (SGPT) </=3 x upper limit of normal; serum bilirubin and alkaline phosphatase </=2 x upper limit of normal, unless proven to be due to disease involvement.
5. Adequate pulmonary function with FEV1, FVC and DLCO >/=50% of expected corrected for hemoglobin and/or volume.
6. Adequate cardiac function with left ventricular ejection fraction >/=40%. No uncontrolled arrhythmias or symptomatic cardiac disease.
7. Zubrod performance status <2.
8. Negative Beta HCG text in a woman with child-bearing potential, defined as not post-menopausal for 12 months or no previous surgical sterilization.

Exclusion Criteria:

1. Patients with grade >/= 3 non-hematologic toxicity from previous therapy that has not resolved to </= grade 1.
2. Prior whole brain irradiation.
3. Active hepatitis B, either active carrier (HBsAg +) or viremic (HBV DNA >/=10,000 copies/mL, or >/= 2,000 IU/mL).
4. Evidence of either cirrhosis or stage 3-4 liver fibrosis in patients with chronic hepatitis C or positive hepatitis C serology.
5. Active infection requiring parenteral antibiotics.
6. HIV infection, unless the patient is receiving effective antiretroviral therapy with undetectable viral load and normal CD4 counts.
7. Patients having received radiation therapy to head and neck (excluding eyes), and internal organs of chest, abdomen or pelvis in the month prior to enrollment.
8. Autologous stem-cell transplant in the previous six months.