$P_5(w) = c_0 + c_1w + c_2w^2 +...+ c_5w^5$, where $ c_0, ... , c_5 $are integers that I want to determine.

$Q_5(w) = w^2 (\frac{d}{dw}( \frac{P_5(w)}{w}) = -c_0 + c_2w^2 + 2c_3w^3 + 3c_4w^4 +4c_5w^5$, another deg 5 polynomial.

I have that every root (over $\mathbb{C}$) of \begin{equation} (Q_5(w))^2 + 16Q_5(w)(1 + w)^3 w^2 - 80(1 + w)^2 w^3 P_5(w) = 0 \end{equation} is also a root of \begin{equation} (24)^3 (1 + w)^5 w^4 [4P_5 (w) w - Q_5(w) ( 1 - w)]^3 + 108(Q_5(w))^5 = 0 \end{equation}

Is there a way to solve this using an algebra software?