1. **State the problem:** Divide the polynomial $4x^3 - 4x^2 - 16x + 7$ by the product $(x+1)(x-2)$ using long division, then express the result as partial fractions. 2. **Rewrite the divisor:** Note that $(x+1)(x-2) = x^2 - x - 2$. 3. **Perform long division:** Divide $4x^3 - 4x^2 - 16x + 7$ by $x^2 - x - 2$. - First term: $\frac{4x^3}{x^2} = 4x$. - Multiply divisor by $4x$: $4x(x^2 - x - 2) = 4x^3 - 4x^2 - 8x$. - Subtract: $$\left(4x^3 - 4x^2 - 16x + 7\right) - \left(4x^3 - 4x^2 - 8x\right) = -8x + 7$$ 4. **Express remainder over divisor:** The division gives quotient $4x$ and remainder $-8x + 7$. So, $$\frac{4x^3 - 4x^2 - 16x + 7}{(x+1)(x-2)} = 4x + \frac{-8x + 7}{(x+1)(x-2)}$$ 5. **Express the remainder fraction in partial fractions:** Assume $$\frac{-8x + 7}{(x+1)(x-2)} = \frac{A}{x+1} + \frac{B}{x-2}$$ Multiply both sides by $(x+1)(x-2)$: $$-8x + 7 = A(x-2) + B(x+1)$$ 6. **Find coefficients A and B:** Set $x = 2$: $$-8(2) + 7 = A(0) + B(3) \Rightarrow -16 + 7 = 3B \Rightarrow -9 = 3B \Rightarrow B = -3$$ Set $x = -1$: $$-8(-1) + 7 = A(-3) + B(0) \Rightarrow 8 + 7 = -3A \Rightarrow 15 = -3A \Rightarrow A = -5$$ 7. **Write the final partial fraction decomposition:** $$\frac{4x^3 - 4x^2 - 16x + 7}{(x+1)(x-2)} = 4x - \frac{5}{x+1} - \frac{3}{x-2}$$ **Final answer:** $$\boxed{4x - \frac{5}{x+1} - \frac{3}{x-2}}$$