Answer
$L(x)=2x$ at $x=x_{0}$
Work Step by Step
$f(x)={x}^2+2x $,
$L(x)=f(a)+f'(a)(x-a)$
$f'(x)=2{x}+2$
$x_{0}=0$
$f(0)=0+0=0$
$f'(0)=0+2=2$
then $L(x)=0+2(x-0)=2x$
$L(x)=2x$ at $x=x_{0}$
Thus, the final answer is: $L(x)=2x$ at $x=x_{0}$