Explanation

(d)
$\frac{1 + 2x}{(1 - 2x)^{2}} = (1 + 2x)(1 - 2x)^{- 2}$
$= (1 + 2x)\left( 1 + \frac{2}{1!}(2x) + \frac{2 \bullet 3}{2!}(2x)^{2} + \ldots + \frac{2 \bullet 3\ldots r}{(r - 1)!}(2x)^{r} + \frac{2 \bullet 3 \bullet 4\ldots(r + 1)(2x)^{r}}{r!} \right)$
$The\ coefficient\ of\ x^{r} = 2\frac{r!}{(r - 1)!}2^{r - 1} + \frac{(r + 1)!}{r!}2^{r}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = {r2}^{r} + (r + 1)r^{2}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 2^{r}(2r + 1)$