Use the Factor Theorem to determine whether \( x+2 \) is a factor of \( P(x)=-2 x^{3}-3 x^{2}-4 x+7 \) Specifically, evaluate \( P \) at the proper value, and then determine whether \( x+2 \) is a factor. \( \begin{array}{l}P(\bar{I})=\square \\ x+2 \text { is a factor of } P(x) \\ x+2 \text { is not a factor of } P(x)\end{array} \)

To determine whether \( x + 2 \) is a factor of \( P(x) = -2x^3 - 3x^2 - 4x + 7 \) using the Factor Theorem, we need to evaluate \( P \) at the value that makes \( x + 2 = 0 \). This occurs when \( x = -2 \). Now, we will calculate \( P(-2) \): \[ P(-2) = -2(-2)^3 - 3(-2)^2 - 4(-2) + 7 \] Calculating each term step-by-step: 1. \( -2(-2)^3 = -2(-8) = 16 \) 2. \( -3(-2)^2 = -3(4) = -12 \) 3. \( -4(-2) = 8 \) 4. The constant term is \( +7 \) Now, we combine these results: \[ P(-2) = 16 - 12 + 8 + 7 \] Calculating this step-by-step: \[ P(-2) = 16 - 12 = 4 \] \[ P(-2) = 4 + 8 = 12 \] \[ P(-2) = 12 + 7 = 19 \] Thus, we find: \[ P(-2) = 19 \] Since \( P(-2) \neq 0 \), according to the Factor Theorem, \( x + 2 \) is not a factor of \( P(x) \). In conclusion: \[ \begin{array}{l} P(-2) = 19 \\ x + 2 \text{ is not a factor of } P(x) \end{array} \]