1. **State the problem:** Solve the equation $ (x+1)(2x+5) = (2x+3)(x-4) + 5 $. 2. **Use the distributive property (FOIL) to expand both sides:** $$ (x+1)(2x+5) = 2x^2 + 5x + 2x + 5 = 2x^2 + 7x + 5 $$ $$ (2x+3)(x-4) = 2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12 $$ 3. **Rewrite the equation with expansions:** $$ 2x^2 + 7x + 5 = 2x^2 - 5x - 12 + 5 $$ 4. **Simplify the right side:** $$ 2x^2 + 7x + 5 = 2x^2 - 5x - 7 $$ 5. **Subtract $2x^2$ from both sides to eliminate quadratic terms:** $$ \cancel{2x^2} + 7x + 5 = \cancel{2x^2} - 5x - 7 $$ $$ 7x + 5 = -5x - 7 $$ 6. **Add $5x$ to both sides to collect $x$ terms on one side:** $$ 7x + 5x + 5 = -5x + 5x - 7 $$ $$ 12x + 5 = -7 $$ 7. **Subtract 5 from both sides:** $$ 12x + \cancel{5} - \cancel{5} = -7 - 5 $$ $$ 12x = -12 $$ 8. **Divide both sides by 12:** $$ x = \frac{-12}{12} $$ $$ x = -1 $$ **Final answer:** $x = -1$