Answer
$ a.\qquad (f\circ g)(x)=x^{4}+8x^{2}+16,\qquad$ Domain: $\mathbb{R}$
$ b.\qquad (g\circ f)(x)=x^{4}+4,\qquad$ Domain: $\mathbb{R}$
$ c.\qquad (f\circ f)(x)=x^{4},\qquad$ Domain: $\mathbb{R}$
$ d.\qquad(g\circ g)(x)=x^{4}+8x^{2}+20,\qquad$ Domain: $\mathbb{R}$
Work Step by Step
$ f(x)=x^{2},\qquad$ Domain: $\mathbb{R}$
$ g(x)=x^{2}+4,\qquad$ Domain: $\mathbb{R}$
$a.$
$f\circ g(x)=f[g(x)]=[g(x)]^{2}$
$=(x^{2}+4)^{2}$
$=x^{4}+8x^{2}+16,\qquad$ Domain: $\mathbb{R}$
$b.$
$(g\circ f)(x)=g[f(x)] =[f(x)]^{2}+4$
$=(x^{2})^{2}+4$
$=x^{4}+4,\qquad$ Domain: $\mathbb{R}$
$c.$
$f\circ f(x)=f[f(x)]=[f(x)]^{2}$
$=(x^{2})^{2}$
$=x^{4},\qquad$ Domain: $\mathbb{R}$
$d.$
$(g\circ g)(x)=g[g(x)] =[g(x)]^{2}+4$
$=(x^{2}+4)^{2}+4$
$=x^{4}+8x^{2}+16+4$
$=x^{4}+8x^{2}+20,\qquad$ Domain: $\mathbb{R}$