Question:

Step 1: Understand the factor theorem. If $(x+1)$ is a factor of the polynomial $p(x) = x^3 + kx^2 - 3x + k + 2$, then by the Factor Theorem, $x = -1$ must be a root of the polynomial. This means $p(-1) = 0$. Step 2: Substitute $x = -1$ into the polynomial. We substitute $x = -1$ into the expression: $$p(-1) = (-1)^3 + k(-1)^2 - 3(-1) + k + 2$$ Step 3: Simplify each term. Let us evaluate each term: - $(-1)^3 = -1$ - $k(-1)^2 = k(1) = k$ - $-3(-1) = 3$ - The constant terms are $k + 2$ So we have: $$p(-1) = -1 + k + 3 + k + 2$$ Step 4: Combine like terms. Collecting all terms: $$p(-1) = -1 + 3 + 2 + k + k = 4 + 2k$$ Step 5: Set the polynomial equal to zero and solve for $k$. Since $(x+1)$ is a factor, we must have $p(-1) = 0$: $$4 + 2k = 0$$ $$2k = -4$$ $$k = -2$$ Step 6: State the final answer. The value of $k$ is $\boxed{-2}$, which corresponds to **Option 3**.