Chapter 10 - The Laplace Transform and Some Elementary Applications - 10.1 Definition of the Laplace Transform - Problems - Page 675: 14

$$\dfrac{2s-4-s^2}{s^2(s-2)}$$

The Laplace Transform can be written as: $L[F(t)]=\int_{0}^{\infty} e^{-st} f(t) dt $ We are given that $f(t)=2 t-e^{2t}$ Now, $L[F(t)]=\int_{0}^{\infty} e^{-st} f(t) dt \\=\int_{0}^{\infty} e^{-st} [2 t-e^{2t}] dt\\=[\dfrac{2}{s^2}]- [\dfrac{1}{s-2}] \\=\dfrac{2s-4-s^2}{s^2(s-2)}$