Use factoring to solve the polynomial equation. \[ x^{3}+8 x^{2}-4 x-32=0 \]

To solve the polynomial equation \( x^{3}+8 x^{2}-4 x-32=0 \) by factoring, we can first try to find rational roots using the Rational Root Theorem. We can test possible rational roots such as \( \pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \pm 32 \). After testing, we find that \( x = 2 \) is a root. Now we can use synthetic division to divide the polynomial by \( x - 2 \): 1. Set up the synthetic division with the coefficients (1, 8, -4, -32). 2. Bring down the 1, multiply by 2, add down: - \( 1 \) - \( 8 + 2 = 10 \) - \( -4 + 20 = 16 \) - \( -32 + 32 = 0 \) The result gives us \( x^2 + 10x + 16 \). Next, we can factor \( x^2 + 10x + 16 \) into \( (x + 2)(x + 8) \). Thus, we have factored the original equation as: \[ (x - 2)(x + 2)(x + 8) = 0 \] Setting each factor equal to zero gives the solutions: \[ x - 2 = 0 \implies x = 2 \] \[ x + 2 = 0 \implies x = -2 \] \[ x + 8 = 0 \implies x = -8 \] The solutions to the equation are \( x = 2, -2, -8 \).