Explanation

(a)
Let $I = \int_{}^{}{\left( \frac{x + 2}{x + 4} \right)^{2}e^{x}dx} = \int_{}^{}e^{x}\left\lbrack \frac{x^{2} + 4 + 4x}{(x + 4)^{2}} \right\rbrack dx$
$\Rightarrow I = \int_{}^{}e^{x}\left\lbrack \frac{x(x + 4)}{(x + 4)^{2}} + \frac{4}{(x + 4)^{2}} \right\rbrack dx$
$\Rightarrow I = \int_{}^{}\frac{e^{x}\ x}{x + 4}\ dx + \int_{}^{}\frac{4e^{x}}{(x + 4)^{2}}\ dx$
$\Rightarrow I = e^{x}\left( \frac{x}{x + 4} \right) - \int_{}^{}\frac{4e^{x}}{(x + 4)^{2}}\ dx + \int_{}^{}\frac{4e^{x}}{(x + 4)^{2}}\ dx$
$\Rightarrow I = \frac{xe^{x}}{(x + 4)} + c$