JEE Math Practice Question

For a polynomial $g(x)$ with real coefficients, let $m_{g}$ denote the number of distinct real roots of $g(x)$. Suppose $S$ is the set of polynomials with real coefficients defined by \[ S=\left\{\left(x^{2}-1\right)^{2}\left(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}\right): a_{0}, a_{1}, a_{2}, a_{3} \in \mathbb{R}\right\} \] For a polynomial $f$, let $f^{\prime}$ and $f^{\prime \prime}$ denote its first and second order derivatives, respectively. Then what is the minimum possible value of $\left(m_{f^{\prime}}+m_{f^{\prime \prime}}\right)$, where $f \in S$?