The function \( f \) and \( g \) are such that \( f(x)=3 x-1 \) and \( g(x) \) \( =x^{2}+4 \). Given that \( f g(x)=2 g f(x) \), what is the value of \( 15 x^{2}-12 x-1 ? \) Select one: a. -1 b. \( \frac{1}{2} \) c. 0 d. 1

To solve for \( 15x^2 - 12x - 1 \) based on the relationship \( fg(x) = 2gf(x) \), we start by calculating both sides of the equation: 1. \( fg(x) = f(g(x)) = f(x^2 + 4) = 3(x^2 + 4) - 1 = 3x^2 + 12 - 1 = 3x^2 + 11 \). 2. Now for \( gf(x) = g(f(x)) = g(3x - 1) = (3x - 1)^2 + 4 = 9x^2 - 6x + 1 + 4 = 9x^2 - 6x + 5 \). 3. We set \( fg(x) = 2gf(x) \): \[ 3x^2 + 11 = 2(9x^2 - 6x + 5). \] 4. Simplifying the right side: \[ 3x^2 + 11 = 18x^2 - 12x + 10. \] 5. Bring all terms to one side: \[ 0 = 18x^2 - 12x + 10 - 3x^2 - 11 = 15x^2 - 12x - 1. \] Now, we see that \( 15x^2 - 12x - 1 = 0 \). Thus, the value of \( 15x^2 - 12x - 1 \) simplifies to zero: The answer is \( \boxed{0} \).