(19) Let $\alpha, \beta$ be the two real roots of the quadratic equation $x^{2}-2 k x+k+20=0$. Find the minimum value of $(\alpha+1)^{2}+$ $(\beta+1)^{2}$, and indicate the value of $k$ when the minimum value is achieved.
Given that $\alpha, \beta$ are the two real roots of the quadratic equation $x^{2}-2 k x+k+20=0$, we have $\Delta=4 k^{2}-$ $4(k+20) \geqslant 0$, which simplifies to $k^{2}-k-20 \geqslant 0$. Solving this, we get $k \geqslant 5$ or $k \leqslant-4$. Additionally, $(\alpha+1)^{2}+(\beta+1)^{2}=$ $(\alpha+\beta)^{2}-2 \alpha \beta+2(\alpha+\beta)+2=4 k^{2}+2 k-38=4\left(k+\frac{1}{4}\right)^{2}-\frac{1}{4}-38$, so when $k=-4$, $(\alpha+1)^{2}+(\beta+1)^{2}$ achieves its minimum value of 18.