2. Determine all values of the real parameter $k$ such that:

$$
(\forall x \in \mathbb{R})\left(k x^{2}-4 k x+k^{2}+2 k-3>0\right)
$$
2. For the given expression to be positive for every real number $x$, the coefficient of the leading term must be positive, i.e., $k>0$, and the discriminant must be negative, i.e.,

$0>(4 k)^{2}-4 k\left(k^{2}+2 k-3\right)=-4 k^{3}+8 k^{2}+12 k=-4 k\left(k^{2}-2 k-3\right)=-4 k(k+1)(k-3)$.

Considering the earlier condition $k>0$, this reduces to $(k+1)(k-3)>0$, i.e., $k \in(-\infty,-1) \cup(3, \infty)$. Since we are only interested in positive values of $k$, the final solution to the problem is $k \in(3, \infty)$.