In order to transform the series we need one additional technique besides differentiation/integration.
We obtain
\begin{align*}
\color{blue}{\sum_{n=0}^{\infty}}&\color{blue}{\frac{(-1)^{n+1}(2n+2)x^{2n+1}}{(2n)!}}\\
&=\frac{d}{dx}\left(\sum_{n=0}^{\infty}(-1)^{n+1}\frac{x^{2n+2}}{(2n)!}\right)\\
&=-\frac{d}{dx}\left(x^2\sum_{n=0}^{\infty}\frac{(ix)^{2n}}{(2n)!}\right)\\
&=-\frac{d}{dx}\left(x^2\sum_{n=0}^{\infty}\color{blue}{\frac{1+(-1)^n}{2}}\,\frac{(ix)^n}{n!}\right)\tag{*}\\
&=-\frac{d}{dx}\left(\frac{x^2}{2}\sum_{n=0}^{\infty}\frac{(ix)^n}{n}+\frac{x^2}{2}\sum_{n=0}^{\infty}\frac{(-ix)^n}{n!}\right)\\
&=-\frac{d}{dx}\left(x^2\,\frac{e^{ix}+e^{-ix}}{2}\right)\\
&=-\frac{d}{dx}\left(x^2\cos(x)\right)\\
&\,\,\color{blue}{=-2x\cos(x)+x^2\sin(x)}
\end{align*}
We observe besides knowing the series expansion of some standard functions like $e^x, \sin(x), \cos(x)$ we need as additional technique besides differentiation the so-called multisection (*) of series. See for instance this MSE post for additional info. Here we have the simplest form of multisection, namely subdivision in even and odd parts via
\begin{align*}
\frac{1+(-1)^n}{2}
\end{align*}
Hint:
A collection of useful techniques can be found when going through chapter 2 of generatingfunctionology by H. S. Wilf.
Section 2.5 Some useful power series contains a list of series expansions which is helpful to keep in mind.
Let's consider two series
\begin{align*}
A(x)=\sum_{n=0}^{\infty}a_nx^n\qquad\qquad B(x)=\sum_{n=0}^{\infty}b_n\frac{x^n}{n!}
\end{align*}
Two relations which are sometimes useful and which can also be found in the cited book are
\begin{align*}
&\left(a_n\right)_{n\geq 0}\quad\to\quad \left(\sum_{k=0}^{n}a_k\right)_{n\geq 0}&&A(x)\quad\to\quad \frac{1}{1-x}A(x)\\
&\left(b_n\right)_{n\geq 0}\quad\to\quad \left(\sum_{k=0}^{n}\binom{n}{k}b_k\right)_{n\geq 0}&&B(x)\quad\to\quad e^xB(x)\\
\end{align*}
