Mistake in simplification of large polynomial inequality? We are to solve for $p$, and the inequality to simplify is 
$$10p^3(1-p)^2 + 5p^4(1-p) + p^5 - 3p^2(1-p) - p^3 > 0$$
On the next line of the textbook, the author simplifies this expression to 
$$3(p-1)^2(2p-1) > 0 \implies p > \frac{1}{2}$$
Since no work was shown, I attempted to reach the same result but got 
$$3(p-1)^2(2p-1) \cdot \mathbb{p^2} > 0$$
Have I made a mistake in my work, or are the answers possibly equivalent? Might there be any reason to leave out the $p^2$ factor from the inequality I reached?

The original expression factorizes as $4p^2(p-1)(p-3)(p-\frac14)>0.$ So $\frac14<p<1$ or $p>3.$
