1. **State the problem:** We need to find the perimeter of an L-shaped polygon with given side lengths: top horizontal $4x - 6$, left vertical $x + 13$, bottom horizontal $2x + 1$, bottom-right vertical $x + 3$, inner horizontal $x + 1$, and right vertical $2x - 4$. 2. **Recall the perimeter formula:** The perimeter is the sum of all outer side lengths of the polygon. 3. **Identify the outer sides:** For the L-shape, the perimeter includes the top horizontal, left vertical, bottom horizontal, bottom-right vertical, and right vertical sides. The inner horizontal side $x + 1$ is inside the shape and does not contribute to the perimeter. 4. **Sum the outer sides:** $$P = (4x - 6) + (x + 13) + (2x + 1) + (x + 3) + (2x - 4)$$ 5. **Combine like terms:** $$P = 4x - 6 + x + 13 + 2x + 1 + x + 3 + 2x - 4$$ $$P = (4x + x + 2x + x + 2x) + (-6 + 13 + 1 + 3 - 4)$$ $$P = 10x + 7$$ 6. **Final answer:** The perimeter in simplest form is $$\boxed{10x + 7}$$