求 $m$ 的取值范围，使得对于所有实数 $x$，$x^2 - 2mx + 2m$ 都大于 $-3$。

1. Rearrange the inequality into the standard quadratic form

$x^2 - 2mx + 2m > -3$
$x^2 - 2mx + (2m + 3) > 0$

2. Apply the discriminant condition for a quadratic to be always positive

For the quadratic expression to be strictly positive for all real $x$, the coefficient of $x^2$ must be positive (which is $1 > 0$) and the discriminant must be negative ($b^2 - 4ac < 0$).
$(-2m)^2 - 4(1)(2m + 3) < 0$

3. Solve the resulting quadratic inequality for $m$

$4m^2 - 8m - 12 < 0$
Dividing by 4:
$m^2 - 2m - 3 < 0$
$(m - 3)(m + 1) < 0$
$-1 < m < 3$