Let $P=x+y+z+u, Q=xy+xz+xu+yz+yu+zu$. Then the first equation becomes $x^2-(Q-xy-xz-xu)=a$ $x^2-Q+x(P-x)=a$ $Px-Q=a$ Similarly, the other equations become $Py-Q=b$ $Pz-Q=c$ $Pu-Q=d$ Now $x={a+Q\over P},y={b+Q\over P},z={c+Q\over P},u={d+Q\over P}$, hence $P={a+b+c+d+4Q\over P}\implies P^2=4Q+a+b+c+d\quad(*)$ $Q={ab+ac+ad+bc+bd+cd+3Q(a+b+c+d)+6Q^2\over P^2}$ Plugging $(*)$ into this we get $4Q^2+(a+b+c+d)Q=ab+ac+ad+bc+bd+cd+3Q(a+b+c+d)+6Q^2$, hence $2Q^2+2Q(a+b+c+d)+ab+ac+ad+bc+bd+cd=0$ Thus $Q_{1,2}={-(a+b+c+d)\pm\sqrt{a^2+b^2+c^2+d^2}\over 2}$ For each of those values we get two quadruplets: $(x,y,z,u)=\left({a+Q\over\sqrt{4Q+a+b+c+d}},{b+Q\over\sqrt{4Q+a+b+c+d}},{c+Q\over\sqrt{4Q+a+b+c+d}},{d+Q\over\sqrt{4Q+a+b+c+d}}\right)$ or $(x,y,z,u)=\left(-{a+Q\over\sqrt{4Q+a+b+c+d}},-{b+Q\over\sqrt{4Q+a+b+c+d}},-{c+Q\over\sqrt{4Q+a+b+c+d}},-{d+Q\over\sqrt{4Q+a+b+c+d}}\right)$