Question

Find the coefficient of $x^{12}$ in $\frac{4+2x-x^2}{{(1+x)}^3}$

Answer 1

$(4+2x-x^2)(1+x{)}^{-3}$

$\underset{\downarrow }{\underset{x^{10}coeff}{\underset{\downarrow }{4(1+x{)}^{-3}}}}+\underset{x^{9}coeff}{\underset{\downarrow }{2x(1+x{)}^{-3}}}-\underset{x^{8}coeff}{\underset{\downarrow }{x^2(1+x{)}^{-3}}}$

$\frac{(-3)(-3-1)(-3-2)(-3-3)(-3-4)(-3-5)(-3-6)(-3-7)(-3-8)(-3-9)}{10!}+$

$\frac{2(-3)(-3-1)(-3-2)(-3-3)(-3-4)(-3-5)(-3-6)(-3-7)(-3-8)}{9!}-$ $\frac{(-3)(-3-1)(-3-2)(-3-3)(-3-4)(-3-5)(-3-6)(-3-7)}{8!}$

$\frac{\left(4\right)(+3)(+4)(+5)(+6)(+7)(+8)(+9)(+10)(+11)(+12)}{10!}+$ $\frac{2(+3)(+4)(+5)(+6)(+7)(+8)(+9)(+10)(-11)}{9!}-$ $\frac{(+3)(+4)(+5)(+6)(+7)(+8)(+9)(+10)}{8!}$

$\frac{4\times 3\times 4\times 5\times 6\times 7\times 8\times 9\times 10\times 11\times 12}{10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}-\frac{2\times 3\times 4\times 5\times 6\times 7\times 8\times 9\times 10\times 11}{9\times 8\times 7\times 6\times 5\times 4\times 3\times 2}-$ $\frac{3\times 4\times 5\times 6\times 7\times 8\times 9\times 10}{8\times 7\times 6\times 5\times 4\times 3\times 2}$

$2\times 11\times 12-10\times 11-9\times 5=264-110-45$

$=109$