1. **State the problem:** We need to find the length of segment $PQ$ in a right triangle where sides $PS = x + 2$ and $PR = 2x - 1$. 2. **Identify the relationship:** Since $PQ$ is the hypotenuse of the right triangle with legs $PS$ and $PR$, we use the Pythagorean theorem: $$PQ^2 = PS^2 + PR^2$$ 3. **Write the formula with given expressions:** $$PQ^2 = (x + 2)^2 + (2x - 1)^2$$ 4. **Expand the squares:** $$(x + 2)^2 = x^2 + 4x + 4$$ $$(2x - 1)^2 = 4x^2 - 4x + 1$$ 5. **Add the expanded terms:** $$PQ^2 = (x^2 + 4x + 4) + (4x^2 - 4x + 1) = 5x^2 + 5$$ 6. **Simplify the expression:** $$PQ^2 = 5x^2 + 5 = 5(x^2 + 1)$$ 7. **Take the square root to find $PQ$:** $$PQ = \sqrt{5(x^2 + 1)} = \sqrt{5} \cdot \sqrt{x^2 + 1}$$ **Final answer:** $$PQ = \sqrt{5} \cdot \sqrt{x^2 + 1}$$ This is the length of $PQ$ in terms of $x$.