已知 $m^{2}=3 n+a, n^{2}=3 m+a, m \neq n$, 则 $m^{2}+2 m n+n^{2}$ 的值为 ( )
$\because m^{2}=3 n+a, n^{2}=3 m+a$,

$\therefore m^{2}-n^{2}=3 n-3 m$,

$\therefore(m+n)(m-n)+3(m-n)=0$,

$\therefore(m-n)[(m+n)+3]=0$,

$\because m \neq n$,

$\therefore(m+n)+3=0$,

$\therefore m+n=-3$,

$\therefore m^{2}+2 m n+n^{2}=(m+n)^{2}=(-3)^{2}=9$.