1. **State the problem:** Simplify and solve the equation $ (x + 1)(x + 4) + 3(x - 2)(x - 1) = 4x(x - 6) $. 2. **Use the distributive property (FOIL) to expand each product:** $$ (x + 1)(x + 4) = x^2 + 4x + x + 4 = x^2 + 5x + 4 $$ $$ (x - 2)(x - 1) = x^2 - x - 2x + 2 = x^2 - 3x + 2 $$ $$ 4x(x - 6) = 4x^2 - 24x $$ 3. **Substitute the expanded forms back into the equation:** $$ x^2 + 5x + 4 + 3(x^2 - 3x + 2) = 4x^2 - 24x $$ 4. **Distribute the 3 in the second term:** $$ x^2 + 5x + 4 + 3x^2 - 9x + 6 = 4x^2 - 24x $$ 5. **Combine like terms on the left side:** $$ (x^2 + 3x^2) + (5x - 9x) + (4 + 6) = 4x^2 - 24x $$ $$ 4x^2 - 4x + 10 = 4x^2 - 24x $$ 6. **Subtract $4x^2$ from both sides:** $$ \cancel{4x^2} - 4x + 10 = \cancel{4x^2} - 24x $$ $$ -4x + 10 = -24x $$ 7. **Add $4x$ to both sides:** $$ -4x + 4x + 10 = -24x + 4x $$ $$ 10 = -20x $$ 8. **Divide both sides by $-20$ to solve for $x$:** $$ x = \frac{10}{-20} = -\frac{1}{2} $$ **Final answer:** $x = -\frac{1}{2}$