Statement-1: Coefficient of $x^n$ in $\sum _{j=0}^n (-1{)}^j(2x+3{)}^{n-j}(5-2x{)}^j$ is $(n+1)2^n$

Statement-2: $\sum _{j=0}^n a^{n-j}{b}^j=\frac{a^{n+1}-b^{n+1}}{a-b},a\ne b$

and coefficient of $x^m$in $(\alpha x+\beta {)}^{m+1}$is $(m+1){\alpha }^m\beta$

$\sum _{i=0}^n a^{n-j}{b}^j=a^n+a^{n-1}b+a^{n-2}{b}^2+\dots +$

              $a{b}^{n-1}+b^n$

$=\frac{a^n\left[1-(b/a{)}^{n+1}\right]}{1-b/a}$                   [sum of a G.P.]

        $=\frac{a^{n+1}-b^{n+1}}{a-b}$

Also, coefficient of $x^m$ in $(\alpha x+\beta {)}^{m+1}$ is $(m+1){\alpha }^m\beta$

$\therefore$ Statement-2 is true.

Next $\sum _{j=0}^n (-1{)}^j(2x+3{)}^{n-j}(5-2x{)}^j$

   $\begin{array}{l}=\sum _{j=0}^n (2x+3{)}^{n-j}(2x-5{)}^j\\ =\frac{(2x+3{)}^{n+1}-(2x-5{)}^{n+1}}{(2x+3)-(2x-5)}\\ =\frac{1}{8}\left[(2x+3{)}^{n+1}-(2x-5{)}^{n+1}\right]\end{array}$

Coefficient of $x^n$ is

$\begin{array}{l}\frac{1}{8}\left[{ }^{n+1}{C}_n{2}^n\left(3\right){-}^{n+1}{C}_n{2}^n(-5)\right]\\ =(n+1)2^n\end{array}$

$\therefore$ Statement-1 is true and Statement-2 is correct explanation for it.