1. **State the problem:** Simplify the algebraic expression $$\frac{2p}{2p+2} + \frac{7}{5p-20}$$. 2. **Identify common factors and factor denominators:** - Factor the denominator $2p+2$ as $2(p+1)$. - Factor the denominator $5p-20$ as $5(p-4)$. 3. **Rewrite the expression with factored denominators:** $$\frac{2p}{2(p+1)} + \frac{7}{5(p-4)}$$ 4. **Simplify the first fraction by canceling common factors:** $$\frac{\cancel{2}p}{\cancel{2}(p+1)} = \frac{p}{p+1}$$ 5. **Find the least common denominator (LCD):** The LCD is $$5(p+1)(p-4)$$. 6. **Rewrite each fraction with the LCD as denominator:** $$\frac{p}{p+1} = \frac{p \cdot 5(p-4)}{5(p+1)(p-4)} = \frac{5p(p-4)}{5(p+1)(p-4)}$$ $$\frac{7}{5(p-4)} = \frac{7 \cdot (p+1)}{5(p+1)(p-4)} = \frac{7(p+1)}{5(p+1)(p-4)}$$ 7. **Add the numerators over the common denominator:** $$\frac{5p(p-4) + 7(p+1)}{5(p+1)(p-4)}$$ 8. **Expand the numerators:** $$5p^2 - 20p + 7p + 7 = 5p^2 - 13p + 7$$ 9. **Final simplified expression:** $$\frac{5p^2 - 13p + 7}{5(p+1)(p-4)}$$ **Answer:** $$\boxed{\frac{5p^2 - 13p + 7}{5(p+1)(p-4)}}$$