smc

Expand binomials, combine like terms, and subtract terms from both sides. $$x^2+2xm+m^2-x^2-2xn-n^2=m^2-2mn+n^2$$ $$2xm+m^2-2xn-n^2=m^2-2mn+n^2$$ $$2xm-2xn-n^2=-2mn+n^2$$ Get all the x-terms on one side and factor to solve for x. $$2xm-2xn=-2mn+2n^2$$ $$2x(m-n)=-2n(m-n)$$ Since $m\ne n$, both sides can be divided by $m-n$. $$x=-n$$ That means $a=0$ and $b=-1$, so the answer is $\boxed{a=0,b\text{has a unique non-zero value}}$.