1. **State the problem:** We are given a parallelogram PQSR with side lengths labeled as $PQ = (2x + 5)$ cm and $RS = (4x + 1)$ cm. We need to find the length of side $PQ$. 2. **Recall the property of parallelograms:** Opposite sides of a parallelogram are equal in length. Therefore, $PQ = RS$. 3. **Set up the equation:** $$2x + 5 = 4x + 1$$ 4. **Solve for $x$:** Subtract $2x$ from both sides: $$\cancel{2x} + 5 = \cancel{2x} + 4x + 1 \implies 5 = 2x + 1$$ Subtract 1 from both sides: $$5 - 1 = 2x + \cancel{1} - \cancel{1} \implies 4 = 2x$$ Divide both sides by 2: $$\frac{4}{\cancel{2}} = \frac{2x}{\cancel{2}} \implies 2 = x$$ 5. **Find $PQ$ by substituting $x=2$ into $PQ = 2x + 5$:** $$PQ = 2(2) + 5 = 4 + 5 = 9$$ **Final answer:** $PQ = 9$ cm.