Absolute values are not easy to deal with in practice, so start by noting that any solution to $|x^2-3x+2|=mx$ is either a solution to $$x^2-3x+2=mx$$ or a solution to $$x^2-3x+2=-mx\rm .$$
I am sure you will be able to find the non-negative values of $m$ for which the first equation has two real solutions, and also the non-negative values of $m$ for which the second equation has two solutions. The only thing you finally need to ensure, to satisfy the question, is that 2 solutions + 2 solutions = 4 solutions. And that (as you will see) is why the inequality is written as $0<m$, not allowing equality.