The function \( f \) and \( g \) are such that \( f(x)=3 x-1 \) and \( g(x)=x^{2}+4 \). Given that \( f(\mathbf{g}(\mathbf{x})=2 \boldsymbol{g}(\mathbf{x}) \), what is the value of \( 15 x^{2}-12 x-1 ? \)

To solve for \( x \) given the equation \( f(g(x)) = 2g(x) \), we first need to compute \( f(g(x)) \). We know: - \( g(x) = x^2 + 4 \) - \( f(x) = 3x - 1 \) Now, we substitute \( g(x) \) into \( f \): \[ f(g(x)) = f(x^2 + 4) = 3(x^2 + 4) - 1 = 3x^2 + 12 - 1 = 3x^2 + 11. \] Next, we compute \( 2g(x) \): \[ 2g(x) = 2(x^2 + 4) = 2x^2 + 8. \] Now, we set the two expressions equal to each other: \[ 3x^2 + 11 = 2x^2 + 8. \] To solve for \( x \), we move all terms to one side: \[ 3x^2 - 2x^2 + 11 - 8 = 0, \] which simplifies to: \[ x^2 + 3 = 0. \] This gives: \[ x^2 = -3. \] Since there are no real solutions to this equation, we can conclude that there are no real values for \( x \) that satisfy the initial equation. Now, we need to find the value of \( 15x^2 - 12x - 1 \): Substituting \( x^2 = -3 \) into the expression: \[ 15(-3) - 12x - 1 = -45 - 12x - 1 = -46 - 12x. \] As we cannot determine a real value for \( x \), we can't provide a specific number without knowing the value of \( x \) in the complex case. Thus, the answer is: \[ \text{The expression is: } -46 - 12x \text{ for } x = \pm i\sqrt{3} \text{ (imaginary solutions)}. \]