Inclusion Criteria:

- Patients must have histologically confirmed stage IIIB (with malignant pleural
pericardial effusion) or stage IV non-small cell lung cancer

- No prior chemotherapy for advanced or metastatic disease

- ECOG performance status 0 or 1

- Patients must have measurable disease, defined as at least one lesion that can be
accurately measured in at least one dimension (longest diameter to be recorded) as >=
20 mm with conventional techniques or as >= 10 mm with spiral CT scan

- Life expectancy of greater than 12 weeks

- Leukocytes >= 3,000/mcL

- Absolute neutrophil count >= 1,500/mcL

- Platelets >= 100,000/mcL

- Total bilirubin within normal institutional limits

- AST(SGOT)/ALT(SGPT) =< 2.5 X institutional upper limit of normal

- Creatinine within normal institutional limits OR creatinine clearance >= 60
mL/min/1.73 m^2 for patients with creatinine levels above institutional normal

- Vorinostat can cause fetal harm when administered to a pregnant woman; there are no
adequate and well-controlled studies in pregnant women; if this drug is used during
pregnancy, or if the patient becomes pregnant while taking this drug, the patient
should be apprised of the potential hazard to the fetus

- Ability to understand and the willingness to sign a written informed consent document

Exclusion Criteria:

- Patients who have had chemotherapy or radiotherapy within 3 weeks (6 weeks for
nitrosoureas or mitomycin C) prior to entering the study or those who have not
recovered from adverse events due to agents administered more than 4 weeks earlier

- Patients may not be receiving any other investigational agents

- Patients with untreated brain metastases should be excluded from this clinical trial;
however, patients who have stable brain disease (should be off corticosteroids) at
least 3 weeks after completion of appropriate therapy are eligible

- Prior or current use of valproic acid, a HDAC inhibitor

- Peripheral neuropathy of severity greater than grade 1

- Known history of allergic reactions to paclitaxel

- Prior therapy with paclitaxel

- Inability to take oral medications on a continuous basis

- Uncontrolled intercurrent illness including, but not limited to, ongoing or active
infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac
arrhythmia, or psychiatric illness/social situations that would limit compliance with
study requirements

- Pregnant women are excluded from this study because vorinostat is an HDAC inhibitor
with the potential for teratogenic or abortifacient effects; because there is an
unknown but potential risk for adverse events in nursing infants secondary to
treatment of the mother with vorinostat, breastfeeding should be discontinued if the
mother is treated with vorinostat; these potential risks may also apply to other
agents used in this study

- HIV-positive patients receiving combination antiretroviral therapy are ineligible
because of the potential for pharmacokinetic interactions with vorinostat; in
addition, these patients are at increased risk of lethal infections when treated with
marrow-suppressive therapy; appropriate studies will be undertaken in patients
receiving combination antiretroviral therapy when indicated