For a polynomial $g\left(x\right)$ with real coefficients, let $m_g$ denote the number of distinct real roots of $g\left(x\right)$. Suppose $S$ is the set of polynomials with real coefficients defined by $S=\left\{{\left(x^2-1\right)}^2\left(a_0+a_{1}x+a_2{x}^2+a_3{x}^3\right) :a_0, a_1, a_2, a_3\in R\right\}$. For a polynomial $f$, let $f\text{'}$ and $f^{″}$ denote its first and second order derivatives respectively. Then the minimum possible value of $\left(m_{f^{\prime }}+m_{f^{″}}\right)$, where $f\in S$, is ____

Hence, the correct answer is 5.