Coefficient of $x^4$ in multinomial expansion What is the coefficient of $x^4$ in $(1 + x - 2x^2)^7$? What is a quick way to solve this problem using the binomial theorem (I have not learned multinomial theorem)?

By factorizing the inner expresion,
$$(1+x-2x^{2})^{7}=((1+2x)(1-x))^{7}=(1+2x)^{7}(1-x)^{7}$$
Using the Binomial Expansion on both, we get:
$$\begin{align}(1+x-2x^{2})^{7} &= \left(1 + \binom{7}{1}(2x) + \binom{7}{2}(2x)^2 + \binom{7}{3}(2x)^3 + \binom{7}{4}(2x)^4\right) \\
&\times\left(1 + \binom{7}{1}(-x) + \binom{7}{2}(-x)^2 + \binom{7}{3}(-x)^3 + \binom{7}{4}(-x)^4\right)\\
&=  \left(1 + 14x + 84x^2 + 280x^3 + 560x^4 + ...\right)\\
&\times\left(1 - 7x + 21x^2 - 35x^3 + 35x^4 + ...\right)\\
&= (1\times35 + 14\times-35 + 84\times21 + 280\times-7 + 560\times1)x^4 + ...\\
&= -91x^4 + ... \end{align}$$
