\(\displaystyle{f{{\left({t}\right)}}}={\left({t}-{2}-{2}\right)}{u}{\left({t}-{2}\right)}\)

\(\displaystyle={\left({t}-{2}\right)}{u}{\left({t}-{2}\right)}-{2}{u}{\left({t}-{2}\right)}\)

By the linearly and times shift properties,

\(\displaystyle{F}{\left({x}\right)}={L}{\left[{\left({t}-{2}\right)}{u}{\left({t}-{2}\right)}\right]}-{2}{L}{\left[{u}{\left({t}-{2}\right)}\right]}\)

\(\displaystyle={\frac{{{1}}}{{{s}^{{2}}}}}{e}^{{-{2}{s}}}-{\frac{{{2}}}{{{s}}}}{e}^{{-{2}{s}}}\)

\(\displaystyle={\left({\frac{{{1}}}{{{s}^{{2}}}}}-{\frac{{{2}}}{{{s}}}}\right)}{e}^{{-{2}{s}}}\)

b) We can express g(t) as

\(\displaystyle{g{{\left({t}\right)}}}={2}{e}^{{-{4}{\left({t}-{1}+{1}\right)}}}{u}{\left({t}-{1}\right)}\)

\(\displaystyle={2}{e}^{{-{4}}}{e}^{{-{4}{\left({t}-{1}\right)}}}{u}{\left({t}-{1}\right)}\)

By the time shift property,

\(\displaystyle{G}{\left({s}\right)}={2}{e}^{{-{4}}}{L}{\left[{e}^{{-{4}{\left({t}-{1}\right)}}}{u}{\left({t}-{1}\right)}\right]}\)

\(\displaystyle={2}{e}^{{-{4}}}{\frac{{{e}^{{-{s}}}}}{{{s}+{4}}}}\)

\(\displaystyle={\frac{{{2}{e}^{{-{\left({s}+{4}\right)}}}}}{{{s}+{4}}}}\)