1. **State the problem:** We need to find a polynomial expression for the perimeter of the given L-shaped polygon with sides labeled $x$, $3y$, $y$, $2y$, and $3x$. 2. **Recall the formula for perimeter:** The perimeter of a polygon is the sum of the lengths of all its sides. 3. **Identify all sides:** The polygon has five sides with lengths $x$, $3y$, $y$, $2y$, and $3x$. 4. **Sum the sides:** Add all side lengths to get the perimeter: $$P = x + 3y + y + 2y + 3x$$ 5. **Simplify the polynomial:** Combine like terms: - Combine $x$ and $3x$ to get $4x$ - Combine $3y$, $y$, and $2y$ to get $6y$ So, $$P = 4x + 6y$$ 6. **Final answer:** The polynomial for the perimeter of the shape is $$\boxed{4x + 6y}$$