Recall that
[tex]L_s\left\{t^n\right\} = \dfrac{n!}{s^{n+1}}[/tex]
where [tex]L_s\left\{y(t)\}[/tex] is the Laplace transform of y(t) into the s-domain.
Then you have
[tex]L_s\left\{t+t^2+t^3\right\} = \dfrac{1!}{s^{1+1}} + \dfrac{2!}{s^{2+1}} + \dfrac{3!}{s^{3+1}} = \boxed{\dfrac1{s^2} + \dfrac2{s^3} + \dfrac6{s^4}}[/tex]
