How to find the roots of the second derivative of $ f(x)=x^2(x − 3)^3$?

Question

I am horrid at factoring and I have to find the inflection points of $ f(x)=x^2(x − 3)^3$. So I to find the inflection points I need to set $f'$ equal to $0$ So I have $f'(x)=2x(x-3)^3+x^2\cdot3(x-3)^2$ and after some simplifying I ended with $5x^4-36x^3+81x^2-54x$ But I do not know go to go from there. I know one zero is $\frac{6}{5}$ but I do not know how to reach that. A step by step factorization would be greatly appreciated, thanks!

EDIT: Where the function is increasing and decreasing. Inflection points are later.

Answer 1

$2x(x-3)^3 + 3x^2(x-3)^2 = x(x-3)^2[2(x-3)+3x] = x(x-3)^2(5x-6)$