Let $m,n,p$ be natural number greater than $2$. Consider $$f(x)=(x-p+1)(x-m+1)(x-n+1)-x(2x-m-p+2)$$ We also have $g(x)$ which is obtained by changing $m$ to $m+1$ and $n$ to $n-1$ in $f$, i.e. $$g(x)=(x-p+1)(x-m)(x-n+2)-x(2x-m-p+1)$$
Let $\alpha_1 \ge \alpha_2 \ge \alpha_3$ be the roots of $f$ and $\beta_1 \ge \beta_2 \ge \beta_3$ be the roots of $g$. How do we prove that $\alpha_1 >\beta_1,\alpha_2 < \beta_2$ and $\alpha_3 >\beta_3$.