Find \( a \) if \( (x+2) \) is a factor of \( a x^{3}+4 x^{2}-4 \)

To determine the value of \( a \) such that \( (x+2) \) is a factor of \( a x^{3}+4 x^{2}-4 \), we can use the Factor Theorem. According to the theorem, if \( (x+2) \) is a factor, then substituting \( x = -2 \) into the polynomial should yield zero. Let's substitute \( x = -2 \) into the polynomial \( a x^{3}+4 x^{2}-4 \). \[ P(-2) = a(-2)^{3} + 4(-2)^{2} - 4 \] Calculating each term: \[ = a(-8) + 4(4) - 4 \] \[ = -8a + 16 - 4 \] \[ = -8a + 12 \] Now, we set \( P(-2) \) equal to zero: \[ -8a + 12 = 0 \] Solving for \( a \): \[ -8a = -12 \] \[ a = \frac{-12}{-8} = \frac{12}{8} = \frac{3}{2} \] So, the value of \( a \) is \( \frac{3}{2} \).