The definition of an injective function is $f(x_1)=f(x_2) \implies x_1 = x_2$. I am having trouble understanding at what point into the proof do you give up and conclude that a function is not injective? For example the function $f(x)=3x^3-2x$
$$f(x_1)=f(x_2)$$ $$3x_1^3-2x_1 =3x_2^3-2x_2$$ $$x_1(3x_1^2-2)=x_2(3x_2^2-2)$$
I am pretty sure that I am not allowed to get rid of everything in the parentheses since this function is not injective. At what point in the proof would you say that it is unable to solve for $x_1 = x_2$ and how would you find the values that make this function not injective?