\( \left. \begin{array} { l } { ( g \cdot f ) ( x ) : } \\ { f ( x ) = 4 x } \\ { g ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 2 x - 8 } \end{array} \right. \)

To find the expression for \((g \cdot f)(x)\), we need to multiply the functions \(g(x)\) and \(f(x)\) together. Given: \(f(x) = 4x\) \(g(x) = x^3 + 3x^2 + 2x - 8\) To find \((g \cdot f)(x)\), we multiply \(g(x)\) by \(f(x)\): \((g \cdot f)(x) = g(x) \cdot f(x) = (x^3 + 3x^2 + 2x - 8) \cdot 4x\) Now, we can simplify the expression by multiplying the terms together. Simplify the expression by following steps: - step0: Calculate: \(\left(x^{3}+3x^{2}+2x-8\right)\times 4x\) - step1: Multiply the first two terms: \(4\left(x^{3}+3x^{2}+2x-8\right)x\) - step2: Multiply the terms: \(\left(4x^{3}+12x^{2}+8x-32\right)x\) - step3: Apply the distributive property: \(4x^{3}\times x+12x^{2}\times x+8x\times x-32x\) - step4: Multiply the terms: \(4x^{4}+12x^{3}+8x^{2}-32x\) The expression for \((g \cdot f)(x)\) is \(4x^4 + 12x^3 + 8x^2 - 32x\).