If $\mathcal{L} \left\{ f(t) \right\} = F(s)$, then by change of scale property, $\mathcal{L} \left\{ f(at) \right\} = \dfrac{1}{a} F \left( \dfrac{s}{a} \right)$
$\mathcal{L} \left\{ f(t) \right\} = \dfrac{s^2 - s + 1}{(2s + 1)^2 (s - 1)}$
$\mathcal{L} \left\{ f(2t) \right\} = \dfrac{1}{2} \times \dfrac{(s/2)^2 - (s/2) + 1}{[ \, 2(s/2) + 1 \, ]^2 (s/2 - 1)}$
$\mathcal{L} \left\{ f(2t) \right\} = \dfrac{1}{2} \times \dfrac{(s^2/4) - (s/2) + 1}{(s + 1)^2 (s/2 - 1)}$
$\mathcal{L} \left\{ f(2t) \right\} = \dfrac{1}{2} \times \dfrac{(1/4)(s^2 - 2s + 4)}{(1/2)(s + 1)^2 (s - 2)}$
$\mathcal{L} \left\{ f(2t) \right\} = \dfrac{1}{2} \times \dfrac{s^2 - 2s + 4}{2(s + 1)^2 (s - 2)}$
$\mathcal{L} \left\{ f(2t) \right\} = \dfrac{s^2 - 2s + 4}{4(s + 1)^2 (s - 2)}$ answer
