The quickest way to get the series is if you already know the series for sine:
[tex]\sin6x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^n(6x)^{2n+1}}{(2n+1)!}[/tex]
Then just multiply the series by [tex]x^4[/tex] to get
[tex]f(x)=x^4\sin6x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^n6^{2n+1}x^{2n+5}}{(2n+1)!}[/tex]
The first four nonzero terms are simply the first four terms:
[tex]6x^5-\dfrac{6^3}{3!}x^7+\dfrac{6^5}{5!}x^9-\dfrac{6^7}{7!}x^{11}[/tex]
