## Task A-4.3. (4 points)

Determine the real number $A$, given that in the expansion of the polynomial

$$
\left(1+x^{4}\right)^{12}+A\left(x\left(1-x^{2}\right)^{2}\right)^{12}
$$

the coefficient of $x^{12}$ is 100.
## Solution.

In the first addend, $x^{12}$ appears only in the term $\binom{12}{3} \cdot 1^{9} \cdot\left(x^{4}\right)^{3}$,

so the coefficient of $x^{12}$ is exactly $\binom{12}{3}=\frac{12 \cdot 11 \cdot 10}{3 \cdot 2 \cdot 1}=220$.

1 point

In the expression $A\left(x\left(1-x^{2}\right)^{2}\right)^{12}$, the power $x^{12}$ appears with the coefficient $A$.

1 point

Therefore, we need to determine $A$ such that $220+A=100$, so $A=-120$.

1 point