1. (16 points) Given the equation in $x$
$$
x^{2}-\left(m^{2}+2 m-3\right) x+2(m+1)=0
$$

has two real roots that are opposites of each other.
(1) Find the value of the real number $m$;
(2) If the equation in $x$ $x^{2}-(k+m) x-3 m-k-5=0$ has roots that are all integers, find all real numbers $k$ that satisfy the condition.