Example 4 Given the parabola $y=2 x^{2}-2 k x+\left(k^{2}+2 k+2\right)$ is above the parabola $y=x^{2}+2 k x-$ $2 k^{2}-1$, find the range of values for the constant $k$.

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Given the inequality about $x$
$$
2 x^{2}-2 k x+\left(k^{2}+2 k+2\right)>x^{2}+2 k x-2 k^{2}-1,
$$

holds for any $x \in \mathbf{R}$, which means the solution set of $x^{2}-4 k x+3 k^{2}+2 k+3>0$ is $\mathbf{R}$. Therefore,
$$
\Delta=16 k^{2}-4\left(3 k^{2}+2 k+3\right)<0 \Rightarrow k^{2}-2 k-3<0,
$$

Thus, the range of $k$ is $-1<k<3$.