a. ${{\left( a-\frac{b}{2} \right)}^{4}}$
$=C_{4}^{0}{{a}^{4}}+C_{4}^{1}{{a}^{3}}\left( -\frac{b}{2} \right)+C_{4}^{2}{{a}^{2}}{{\left( -\frac{b}{2} \right)}^{2}}+C_{4}^{3}a{{\left( -\frac{b}{2} \right)}^{3}}+C_{4}^{4}{{\left( -\frac{b}{2} \right)}^{4}}$
$={{a}^{4}}-2{{a}^{3}}b+\frac{3}{2}{{a}^{2}}{{b}^{2}}-\frac{1}{2}a{{b}^{3}}+\frac{1}{16}{{b}^{4}}$
b. $(2{{x}^{2}}+1{{)}^{5}}$
$=C_{5}^{0}{{(2{{x}^{2}})}^{5}}+C_{5}^{1}{{(2{{x}^{2}})}^{4}}.1+C_{5}^{2}{{(2{{x}^{2}})}^{3}}{{.1}^{2}}+C_{5}^{3}{{(2{{x}^{2}})}^{2}}{{.1}^{3}}+C_{5}^{4}2{{x}^{2}}{{.1}^{4}}+C_{5}^{5}{{.1}^{5}}$
$=32{{x}^{7}}+80{{x}^{6}}+80{{x}^{6}}+40{{x}^{4}}+10{{x}^{2}}+1$