Let $a$ be a fixed real number. Define $M(t)$ to be the maximum value of $-x^{2}+2 a x+a$ over all real numbers $x$ with $x \leq t$. Determine a polynomial expression in terms of $a$ that is equal to $M(a-1)+M(a+2)$ for every real number $a$.
Let $f(x)=-x^{2}+2 a x+a$.

Since $(x-a)^{2}=x^{2}-2 a x+a^{2}$, then

$$
-x^{2}+2 a x+a=-\left(x^{2}-2 a x+a^{2}\right)+a^{2}+a=-(x-a)^{2}+a^{2}+a
$$

Therefore, $M(t)$ is the maximum value of $-(x-a)^{2}+a^{2}+a$ over all real numbers $x$ with $x \leq t$. Now the graph of $y=f(x)=-(x-a)^{2}+a^{2}+a$ is a parabola opening downwards with vertex at $\left(a, a^{2}+a\right)$.

Since the parabola opens downwards, then the parabola reaches its maximum at the vertex $\left(a, a^{2}+a\right)$.

Therefore, $f(x)$ is increasing when $xa$.

This means that, when $ta$, then $M(a+2)=f(a)$.

Therefore,

$$
\begin{aligned}
M(a-1)+M(a+2) & =f(a-1)+f(a) \\
& =\left(-((a-1)-a)^{2}+a^{2}+a\right)+\left(-(a-a)^{2}+a^{2}+a\right) \\
& =-1+a^{2}+a-0+a^{2}+a \\
& =2 a^{2}+2 a-1
\end{aligned}
$$

ANSWER: $2 a^{2}+2 a-1$