Start by preheating your oven to 350°F (175°C). Grease and flour a bread loaf pan. For easy removal, you can also line it with parchment paper.

In a stand mixer, cream together the softened butter, granulated sugar and brown sugar. This should take about 2 minutes, and you want the mixture to become light and fluffy.

Incorporate the vegetable oil, large eggs, and vanilla extract into the butter and sugar mixture. Mix until everything is well combined and the mixture is creamy.

In a separate bowl, mash the ripe bananas with a fork. The riper the bananas, the sweeter and more flavorful your banana bread will be.

Add the mashed bananas, grated carrots, and rum raisins to the mixing bowl with the wet ingredients. Stir until all the components are thoroughly combined.

In another bowl, combine the all-purpose flour, baking soda, and salt. Give it a good whisk to ensure even distribution of the dry ingredients. Toss the walnut pieces into the flour mixture to coat them evenly. This step helps prevent the walnuts from sinking to the bottom of the bread during baking.

Gradually add the dry ingredient mixture to the wet ingredients. Mix gently until just combined. Be careful not to overmix; it's okay if there are a few lumps.

Pour the batter into the prepared loaf pan and spread it evenly. Bake in the preheated oven for 50-60 minutes or until a toothpick or knife inserted into the center comes out with just a few moist crumbs attached. The exact baking time may vary, so keep an eye on it in the final minutes.

Allow the banana bread to cool in the pan for about 10 minutes. Then, using the parchment paper for easy lifting, transfer it to a wire rack to cool completely.