Let us say that a polynomial with real coefficients is totally real if all its complex roots are real and distinct. Let $P \in \Bbb R [X]$ be totally real. Is it true that
$$Q(X)=\int_0^XP(t)\,dt+aP(X)$$
is also totally real for all real $a$? If not, is this true if one adds the condition $P(0)=0$? Or some other additional condition?
As an additional question, can one give an upper bound on the roots of $Q$ in terms of those of $P$?
EDIT: Sorry, this is false in general, it is in fact easy to give counterexamples. My question is then: what reasonable additional condition should one add ?
Sorry if this is an elementary question.