Can every polynomial $P(x)$ with integer coefficients we represented in the form $$ P(x) = Q^2(x)R(x), $$ where $Q(x)$ and $R(x)$ are polynomials with integer coefficients such that $R(x)$ has no repeated (complex) roots?
If this is true it should be well-known, then a reference would be helpful.
Intuition: if $\alpha$ is any complex root of $P(x)$ of even multiplicity $2k$, then factor $(x-\alpha)^k$ goes to $Q(x)$. Conversely, if $\alpha$ has odd multiplicity $2k+1$, then $(x-\alpha)^k$ goes to $Q(x)$ while $(x-\alpha)$ goes to $R(x)$. However, will $Q(x)$ and $R(x)$ have integer coefficients?