**Study Summary:** This research study is testing how well a medicine called pegloticase works for people with a type of arthritis called gout. Gout is when too much uric acid builds up in your blood, causing painful swelling, especially in the joints. **Pegloticase** is a medicine that helps lower uric acid levels. In this study, it will be given in two ways: **subcutaneously (SC)**, meaning under the skin, and **intravenously (IV)**, meaning directly into a vein. Both methods will be used with another medicine called **methotrexate (MTX)**, which helps the body better handle pegloticase. The study will check how well uric acid levels stay low over six months. **Key Points:** - **Length**: The study lasts for 6 months, with medicine given every two weeks. - **Eligibility**: Participants must be 18 years or older, have uncontrolled gout, and meet specific health criteria. - **Exclusions**: Those with certain health issues like liver problems, uncontrolled diabetes, or kidney disease may not qualify. Participants will help researchers understand the best way to use pegloticase, but there may be some risks due to side effects or health conditions.