1. **State the problem:** We are given the perimeter of a triangle as the polynomial $3x^2 - 5x + 2$ and two sides as $x^2 + 3x + 5$ and $x^2 - x - 8$. We need to find an expression for the third side. 2. **Formula used:** The perimeter $P$ of a triangle is the sum of its three sides: $$P = ext{side}_1 + ext{side}_2 + ext{side}_3$$ 3. **Set up the equation:** Let the third side be $S$. Then: $$3x^2 - 5x + 2 = (x^2 + 3x + 5) + (x^2 - x - 8) + S$$ 4. **Combine like terms on the right side:** $$(x^2 + 3x + 5) + (x^2 - x - 8) = x^2 + x^2 + 3x - x + 5 - 8 = 2x^2 + 2x - 3$$ 5. **Rewrite the equation:** $$3x^2 - 5x + 2 = 2x^2 + 2x - 3 + S$$ 6. **Isolate $S$:** $$S = (3x^2 - 5x + 2) - (2x^2 + 2x - 3)$$ 7. **Subtract polynomials:** $$S = 3x^2 - 5x + 2 - 2x^2 - 2x + 3 = (3x^2 - 2x^2) + (-5x - 2x) + (2 + 3) = x^2 - 7x + 5$$ **Final answer:** The expression for the third side is $$\boxed{x^2 - 7x + 5}$$