Taylor polynomials practice question 7

Differentiating $e^{2x}\sin x$ three times is slow. Which two Maclaurin polynomials do you already know, and how could you combine them?

Write down the Maclaurin polynomials of $e^{2x}$ (put $u = 2x$ in that of $e^{u}$) and of $\sin x$, each up to $x^{3}$, and multiply them.

An $x^{3}$ term of the product comes from every pair of terms whose powers add to 3. As $\sin x \approx x - \tfrac{x^{3}}{6}$ has no constant or $x^{2}$ term, there are two such pairs.