Differentiate. \[ g(x)=\frac{8 x-9}{3 x+1}+x^{3} \] \[ g^{\prime}(x)= \]

Step 1 ：The given function is a combination of a rational function and a polynomial function. To find the derivative, we need to apply the quotient rule to the rational function and the power rule to the polynomial function.

Step 2 ：The quotient rule is given by: \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{vu' - uv'}{v^2}\) where \(u = 8x - 9\), \(v = 3x + 1\), \(u' = \frac{du}{dx}\), and \(v' = \frac{dv}{dx}\).

Step 3 ：The power rule is given by: \(\frac{d}{dx}\left(x^n\right) = nx^{n-1}\) where \(n = 3\).

Step 4 ：Applying these rules, we find that the derivative of the function \(g(x)=\frac{8 x-9}{3 x+1}+x^{3}\) is \(g^{\prime}(x)=3x^{2}+\frac{8}{3x+1}-\frac{3(8x-9)}{(3x+1)^{2}}\).

Step 5 ：\(\boxed{g^{\prime}(x)=3x^{2}+\frac{8}{3x+1}-\frac{3(8x-9)}{(3x+1)^{2}}}\) is the final answer.