Preheat oven to 350°F. Grease a 9×13-inch baking pan.

In a medium bowl, whisk together flour, baking powder, baking soda, and salt.

In a large bowl, cream butter, granulated sugar, and brown sugar until light and fluffy.

Add eggs one at a time, mixing well after each.

Stir in mashed bananas, sour cream, and vanilla extract until combined.

Gradually add dry ingredients to the wet mixture and mix just until incorporated.

Do not overmix.

Pour batter into prepared pan and spread evenly.

Bake for 30–35 minutes, or until a toothpick inserted in the center comes out clean.

Cool completely before slicing and serving.