$11 \cdot 3$ in the polynomial expansion of $\left(1+2 x-x^{2}\right)^{4}$ the coefficient of $x^{7}$ is
(A) -8.
(B) 12.
(C) 6.
(D) -12.
(E) None of these.
(25th American High School Mathematics Examination, 1974)
[Solution] Since $1+2 x-x^{2}=-(x-1)^{2}+2$, then
$$
\begin{aligned}
& \left(1+2 x-x^{2}\right)^{4} \\
= & {\left[2-(x-1)^{2}\right]^{4}=\left[(x-1)^{2}-2\right]^{4} } \\
= & (x-1)^{8}+C_{4}^{3}(x-1)^{2 \cdot 3}(-2)^{1}+\cdots
\end{aligned}
$$

From the second term onwards, the power of $x$ is $\leqslant 6$,
$\therefore\left(1+2 x-x^{2}\right)^{4}$ The $x^{7}$ term in the expansion is the $x^{7}$ term of $(x-1)^{8}$, which is $C_{8}^{1} x^{7}(-1)^{1}$, so its coefficient is -8.
Therefore, the answer is $(A)$.