Let's say $P(x)$ is a polynomial with degree $k$ then $P'(x)$ is a polynomial with degree $k-1$. What can be said about the roots of this polynomial? $$f(x)=P(x)+(x^3 + x)P'(x) $$ Since $(x^3 + x)$ has complex roots, is it necessary that $f(x)$ will have complex roots? If yes then why? And what can be said about the roots?
P.S coefficients of P and P' are real and P' may may or may not be derivative of P
Assume $k\ge3$. Let $$f(x)=x(x-1)(x-2)\cdots(x-k-1)$$ (or any degree $k+2$ polynomial with all roots real). Use polynomial division by $x^3+x$ to arrive at $f(x)-x^k=(x^3+x)P'(x)+R(x)$ and let $P(x)=x^k+R(x)$- Then $\deg P'=k-1$ and $\deg P=k$ as desired and $P(x)+(x^3+3)P'(x)$ has all roots real.