Obtain the Maclaurin series for $x^2 \cos \left(2 x\right)$ 

$x^2 \cos \left(2 x\right)$$\stackrel{\text{formal series}}{\to }$$\sum _{n\=0}^{\mathrm{\∞}}\frac{{\left(-1\right)}^n{4}^n{x}^{2n\+2}}{\left(2n\right)\!}$

Obtain the Maclauring series for $\cos \left(x\right)$

$\cos \left(x\right)$$\stackrel{\text{formal series}}{\to }$$\sum _{n\=0}^{\mathrm{\∞}}\frac{{\left(-1\right)}^n{x}^{2n}}{\left(2n\right)\!}$

In the series for $\cos \left(x\right)$, replace $x$ with $2 x$ and multiply the series termwise by $x^2$ 

$x^2\cdot \(\genfrac{}{}{0ex}{}{\frac{x^{2 n}}{\left(2 n\right)\!}}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=2 x}\)$ = $\frac{x^2{\left(2x\right)}^{2n}}{\left(2n\right)\!}$$\stackrel{\text{assuming integer}}{\to }$$\frac{x^{2n\+2}{4}^n}{\left(2n\right)\!}$$$