12. (20 points) Let $a \in \mathbf{R}$, and for any real number $b$, we have $\max _{x \in[0,1]}\left|x^{2}+a x+b\right| \geqslant 1$. Find the range of values for $a$.
12. Let $f(x)=x^{2}+a x+b$. Then $|f(0)|=|b|$.
Below, we only need to consider $|b|f\left(-\frac{a}{2}\right)$.
Thus, $f(1)=1+a+b \geqslant 1 \Rightarrow a \geqslant 1$.
(2) When $-2 \leqslant a<0$, for $b=0$ we have
$$
\left.|f(1)|=|1+a|<1,|f|-\frac{a}{2}\right) \left.|=| \frac{-a^{2}}{4} \right\rvert\,<1 \text {. }
$$
(3) When $a \leqslant-2$,
$$
f(1)=1+a+b \leqslant-1 \Rightarrow a \leqslant-3 \text {. }
$$

In summary, $a \geqslant 1$ or $a \leqslant-3$.