1. **State the problem:** Simplify the expression $$\frac{x + 7}{x + 5} - \frac{7x}{x^2 - 3x - 40}$$ to a single simplified fraction. 2. **Factor the denominator:** The second denominator is a quadratic. Factor it: $$x^2 - 3x - 40 = (x - 8)(x + 5)$$ 3. **Rewrite the expression with factored denominators:** $$\frac{x + 7}{x + 5} - \frac{7x}{(x - 8)(x + 5)}$$ 4. **Find the common denominator:** The least common denominator (LCD) is $$ (x + 5)(x - 8) $$. 5. **Rewrite each fraction with the LCD:** - First fraction: multiply numerator and denominator by $$ (x - 8) $$ $$\frac{(x + 7)(x - 8)}{(x + 5)(x - 8)}$$ - Second fraction already has the LCD. 6. **Combine the fractions:** $$\frac{(x + 7)(x - 8) - 7x}{(x + 5)(x - 8)}$$ 7. **Expand the numerator:** $$ (x + 7)(x - 8) = x^2 - 8x + 7x - 56 = x^2 - x - 56 $$ 8. **Subtract $$7x$$:** $$ x^2 - x - 56 - 7x = x^2 - 8x - 56 $$ 9. **Simplify numerator:** $$ x^2 - 8x - 56 $$ 10. **Factor numerator if possible:** $$ x^2 - 8x - 56 = (x - 14)(x + 4) $$ 11. **Write the final simplified expression:** $$\frac{(x - 14)(x + 4)}{(x + 5)(x - 8)}$$ **Answer:** $$\frac{(x - 14)(x + 4)}{(x + 5)(x - 8)}$$