I am wondering for which polynomials $P$ does there exist a polynomial $Q$ such that $P(x)=Q(x)Q'(x)$? I am quite sure a complete characterisation will be very hard, but I'm looking for partial results as well.
Clearly, if $P$ has one term, then $P$ has to have odd degree, since the degree of the right handed side is odd. The following example shows that in fact all such polynomials with positive coefficients work:
We see that $Q(x)=\sqrt{\dfrac{m}{n}}x^n$ gives $Q'(x)=\sqrt{mn}x^{n-1}$ gives $ Q(x)Q'(x)=mx^{2n-1}$. If we require $Q$ to have real coefficients then this only works for $m>0$.
We may also take $Q(x)=\sqrt{\dfrac{m}{n}}x^n+C$, this will give $P(x)=ax^{2n-1}+bx^{n-1}$ for all real $a>0$ and all real $b$. I conjecture this are the only two-term polynomials $P$ that statistify, because if we change $Q$, then the product $Q\cdot Q'$ will probably have too many terms.
It is not that hard: the antiderivative of $P(x)$ should be a square.