Symmetric Matrices & Quadratic Forms · Hard problem

Lagrange's method writes a quadratic form as a sum of squares by completing the square one variable at a time. For $q(x,y,z) = x^2+2y^2+3z^2+4xy+2xz$, completing the square in $x$ first and then in $y$ gives $q = (x+2y+z)^2 + a(y+z)^2 + bz^2$ for some numbers $a,b$. Find $a+b$.