11. (20 points) Given
Figure 1
Function
$$
f(x)=x^{2}+|x+a-1|+(a+1)^{2}
$$

has a minimum value greater than 5. Find the range of values for $a$.
11. From the problem, we know that $f(x)$ is a piecewise quadratic function, i.e.,
$$
f(x)=\left\{\begin{array}{ll}
\left(x-\frac{1}{2}\right)^{2}+a^{2}+a+\frac{7}{4}, & x \leqslant 1-a ; \\
\left(x+\frac{1}{2}\right)^{2}+a^{2}+3 a-\frac{1}{4}, & x \geqslant 1-a .
\end{array}\right.
$$

When $1-a>\frac{1}{2}$, i.e., $a5 \\
\Rightarrow a5 \\
\Rightarrow a \geqslant \frac{3}{2} ;
\end{array}
$$

When $-\frac{1}{2}5 \\
\Rightarrow \frac{\sqrt{6}}{2}<a<\frac{3}{2} .
\end{array}
$$

In summary, $a \in\left(-\infty, \frac{-1-\sqrt{14}}{2}\right) \cup\left(\frac{\sqrt{6}}{2},+\infty\right)$.