Preheat the oven to 350°F (175°C). Grease and flour a 9x5-inch loaf pan.
In a large bowl, combine the flour, baking soda, cinnamon, and salt. Set aside.
In another bowl, whisk together the mashed bananas, grated apple, shredded carrots, vegetable oil, eggs, vanilla extract, granulated sugar, and brown sugar until smooth.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined. Do not overmix.
Fold in the chopped walnuts if using.
Pour the batter into the prepared loaf pan and smooth the top with a spatula.
Bake for 50 minutes or until a toothpick inserted into the center comes out clean.
Allow the bread to cool in the pan for 10 minutes, then transfer to a wire rack to cool completely before slicing.
Notes
This bread stays fresh for up to 3 days at room temperature and tastes even better the next day. Wrap tightly to retain moisture.