Remainder Theorem Question: Deduce the Polynomial Given the Remainder

According to the Remainder Theorem, we have: $$p(m)=m \implies 2m^2+n=m \tag1$$ $$p(n) = n \implies n^2+mn + n =n \implies n(m+n)=0 \tag2$$

From $(2)$, it follows that either $n=0$ or $n=-m$.

If $n=0$, then from $(1)$, it must be that $m=0$ or $m=\frac12$.

If $n=-m$, then from $(1)$, it must be that $m=0$ or $m=1$.

Putting this all together, we have the following solutions: $$(m,n)=(0,0), \,(0,\tfrac12), \, (1,-1)$$

I used the idea that if f(x) has remainder r when divided by (x-a), then f(a)=r