The purpose of this study is to assess the safety and efficacy of MK-8457 + Methotrexate (MTX) in participants with active rheumatoid arthritis (RA) despite MTX therapy. The primary hypothesis is that at least 1 dose of MK-8457 + MTX will be superior to placebo + MTX as measured by the percentage of participants who achieve American College of Rheumatology 20 (ACR 20) response after 12 weeks of treatment.