\[x^4\]
in the expansion of \[(3-x)^6(3+x)^4\]
notice that \[9-x^2=(3-x)(3+x)\]
soWrite the question as
\[(3-x)^2(3-x)^4(3+x)^4=(9-6x+x^2)(9-x^2)\]
.We are only interested in the coefficient of
\[x^4\]
so ignore any powers of \[x\]
higher than \[x^4\]
.\[\begin{equation} \begin{aligned} (3-x)^6(3+x)^4 &= (9-6x_x^2)(9-x^2)^4 \\ &= (9-6x+x^2)({}^4C_09^4(-x^2)^0+{}^4C_1 9^3(-x^2)^1+{}^4C_2 9^2(-x^2)^2+ higher \; powers \; of \; x) \\ &= (9-6x+x^2)(6561-2916x^2+486x^4 + higher \; powers \; of \; x ) \end{aligned} \end{equation}\]
The only contributions to the coefficient of
\[x^4\]
are from \[x^2 \times -2916 x^2\]
and \[9 \times 486x^4=4374x^4\]
. The coefficient is \[-2916+4374=1458\]
.