(x-a)(x-b)=x^2-x(a+b)+ab=x^2+px+q; a+b=-p; ab=q
(x-m)(x-n)=x^2-x(m+n)+mn=x^2+qx+p; m+n=-q; mn=p
a^2+b^2=m^2+n^2; a^2+b^2=a^2+2ab+b^2-2ab=(a+b)^2-2ab: p^2-2q
m^2+n^2=(m+n)^2-2mn=q^2-2p
p^2-2q=q^2-2p; p^2-q^2=2(q-p)=-2(p-q)=(p-q)(p+q). Therefore since p<>q, p+q=-2.