No, that's incorrect, I'm afraid. You need to get the series for not Multiply those two series together to get the result.
The question is:
Use multiplication of Taylor series to find the quartic Taylor polynomial about 0 for the function:
evaluating the coefficients.
For sin x the standard Taylor series about 0 is:
For this can be rearranged to fit the standard Talor series
, which is
...
Multiplying these two together the answer I get is
Does this look along the right lines?
Hmm. We have
as you had before. We don't need more terms because we're only looking for the first four terms. According to your formula there, we have
so
We don't need more terms than these, because the lowest power of x in the sin series is to the first power; hence, the cubic in this series will go to a fourth power.
Both of these are correct. Moving on, then:
Can you see where your mistake is now?