1. **Problem statement:** We are given that when a polynomial $P(x)$ is divided by $x+4$, the quotient is $x^2 - x + 7$ and the remainder is $-5$. We need to find $P(x)$. 2. **Recall polynomial division:** The division algorithm for polynomials states: $$P(x) = (x+4) \times \text{quotient} + \text{remainder}$$ 3. **Plug in the given quotient and remainder:** $$P(x) = (x+4)(x^2 - x + 7) - 5$$ 4. **Expand the product:** $$(x+4)(x^2 - x + 7) = x \cdot (x^2 - x + 7) + 4 \cdot (x^2 - x + 7)$$ $$= x^3 - x^2 + 7x + 4x^2 - 4x + 28$$ 5. **Combine like terms:** $$x^3 + (-x^2 + 4x^2) + (7x - 4x) + 28 = x^3 + 3x^2 + 3x + 28$$ 6. **Add the remainder:** $$P(x) = x^3 + 3x^2 + 3x + 28 - 5 = x^3 + 3x^2 + 3x + 23$$ **Final answer:** $$P(x) = x^3 + 3x^2 + 3x + 23$$