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 the equation \( f g(x) = 2 g f(x) \), we first need to express both sides using the functions \( f(x) \) and \( g(x) \). Calculating the left-hand side: \[ g(x) = x^2 + 4 \] \[ f(g(x)) = f(x^2 + 4) = 3(x^2 + 4) - 1 = 3x^2 + 12 - 1 = 3x^2 + 11 \] Now, substituting into the equation: \[ f(g(x)) = 3x^2 + 11 \] Calculating the right-hand side: \[ f(x) = 3x - 1 \] \[ g(f(x)) = g(3x - 1) = (3x - 1)^2 + 4 = 9x^2 - 6x + 1 + 4 = 9x^2 - 6x + 5 \] \[ 2g(f(x)) = 2(9x^2 - 6x + 5) = 18x^2 - 12x + 10 \] Now setting the left-hand side equal to the right-hand side: \[ 3x^2 + 11 = 18x^2 - 12x + 10 \] Rearranging the equation: \[ 0 = 18x^2 - 12x + 10 - 3x^2 - 11 \] \[ 0 = 15x^2 - 12x - 1 \] Thus, we have the expression \( 15x^2 - 12x - 1 = 0 \). The solutions for this quadratic can be found using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): Here, \( a = 15, b = -12, c = -1 \). Calculating the discriminant: \[ b^2 - 4ac = (-12)^2 - 4(15)(-1) = 144 + 60 = 204 \] Finding the roots: \[ x = \frac{12 \pm \sqrt{204}}{30} = \frac{12 \pm \sqrt{102}}{15} \] However, to find the value of \( 15x^2 - 12x - 1 \), we realize we've established that it is already equal to \( 0 \). Thus, the correct answer is: c. 0