For which values of $m$ I get this for any $x$?

Question

For which values of $m$ I get this for any $x$?

$$ (2m-4)x^2 + (m+1)x -1 > 3x - 2 $$

Answer 1

Hint:

It is equivalent to $$ (2m-4)x^2+(m-2)x+1 > 0\quad\text{for all }x, $$ so it means this is a quadratic polynomial with no real root, and with a positive leading coefficient.

Can you continue?

Answer 2

We see that $m=2$ is valid.

For $m\neq2$ we obtain the following system: $$2m-4>0$$ and $$(m-2)^2-4(2m-4)<0.$$ Can you end it now?