1. **State the problem:** We need to find the perimeter of a right triangle with sides labeled as $12 - 4x$, $7x + 1$, and $x - 3$. 2. **Recall the perimeter formula:** The perimeter $P$ of a triangle is the sum of the lengths of all its sides: $$P = \text{side}_1 + \text{side}_2 + \text{side}_3$$ 3. **Write the expression for the perimeter:** $$P = (12 - 4x) + (7x + 1) + (x - 3)$$ 4. **Combine like terms:** $$P = 12 - 4x + 7x + 1 + x - 3$$ Group the $x$ terms and constants: $$P = ( -4x + 7x + x ) + (12 + 1 - 3)$$ 5. **Simplify the expression:** $$P = (\cancel{-4x} + \cancel{7x} + x) + (12 + 1 - 3) = (4x) + (10)$$ 6. **Final simplified perimeter:** $$P = 4x + 10$$ **Answer:** The perimeter of the triangle is $4x + 10$, which corresponds to option B).