Proving Polynomial Identities

Let's try to factor the expression by ourselves beginning by factoring the greatest common factor $3x$ out. $$\begin{array}{c}3x^3+27x\iff 3x(x^2+9)\end{array}$$ At this step, the student factored the sum of two squares as if it was the difference of two squares.

The sum of two squares cannot be factored further. Therefore the factored form of the expression is $3x(x^2+9).$