  For a given real polynomial $p$ we study the possible number of real roots of
a differential polynomial $H_{\varkappa}[p](x) =
\varkappa\left(p'(x)\right)^2-p(x)p''(x), \varkappa \in \mathbb{R}.$ In the
special case when all real zeros of the polynomial $p$ are simple, and all
roots of its derivative $p'$ are real and simple, the distribution of zeros of
$H_{\varkappa}[p]$ is completely described for each real $\varkappa.$ We also
provide counterexamples to two Boris Shapiro's conjectures about the number of
zeros of the function $H_{\frac{n-1}{n}}[p].$
