1. **State the problem:** Simplify the expression $$\frac{p^2+pq}{p^2-pr} \div \frac{p^2-q^2}{p^2-r^2}$$. 2. **Rewrite division as multiplication:** Dividing by a fraction is the same as multiplying by its reciprocal. $$\frac{p^2+pq}{p^2-pr} \times \frac{p^2-r^2}{p^2-q^2}$$ 3. **Factor all expressions where possible:** - Numerator 1: $$p^2+pq = p(p+q)$$ - Denominator 1: $$p^2-pr = p(p-r)$$ - Numerator 2: $$p^2-r^2 = (p-r)(p+r)$$ (difference of squares) - Denominator 2: $$p^2-q^2 = (p-q)(p+q)$$ (difference of squares) 4. **Substitute factored forms:** $$\frac{p(p+q)}{p(p-r)} \times \frac{(p-r)(p+r)}{(p-q)(p+q)}$$ 5. **Cancel common factors:** - Cancel $p$ in numerator and denominator. - Cancel $(p-r)$ in numerator and denominator. - Cancel $(p+q)$ in numerator and denominator. 6. **Simplify remaining expression:** $$\frac{1}{1} \times \frac{(p+r)}{(p-q)} = \frac{p+r}{p-q}$$ **Final answer:** $$\frac{p+r}{p-q}$$