6. Determine whether \( (x-2) \) is a factor of \( p(x)=x^{3}+2 x^{2}-6 x-4 \). If it is, write \( p(x) \) as a product of two factors.

To determine if \( (x-2) \) is a factor of \( p(x) = x^3 + 2x^2 - 6x - 4 \), we can use the Factor Theorem. According to the theorem, if \( (x-c) \) is a factor of the polynomial \( p(x) \), then \( p(c) = 0 \). Let's evaluate \( p(2) \): \[ p(2) = 2^3 + 2(2^2) - 6(2) - 4 \] \[ = 8 + 8 - 12 - 4 \] \[ = 8 + 8 - 12 - 4 = 0 \] Since \( p(2) = 0 \), we conclude that \( (x-2) \) is indeed a factor of \( p(x) \). Next, we can find the other factors by performing polynomial long division of \( p(x) \) by \( (x-2) \). 1. Divide the leading term: \( x^3 / x = x^2 \). 2. Multiply \( x^2 \) by \( (x-2) \): \( x^2(x-2) = x^3 - 2x^2 \). 3. Subtract to get: \( (x^3 + 2x^2 - 6x - 4) - (x^3 - 2x^2) = 4x^2 - 6x - 4 \). 4. Bring down and repeat: \( 4x^2 / x = 4x \). 5. Multiply \( 4x \) by \( (x-2) \): \( 4x(x-2) = 4x^2 - 8x \). 6. Subtract: \( (4x^2 - 6x) - (4x^2 - 8x) = 2x - 4 \). 7. Divide \( 2x - 4 \) by \( (x-2) \): \( 2 \). So, we have: \[ p(x) = (x-2)(x^2 + 4x + 2) \] Thus, the final factored form of the polynomial is: \[ p(x) = (x-2)(x^2 + 4x + 2) \]