Some extensions of Bairstow's method

u _xx +u _yy =u _t Bairstow's method for improving an approximate real quadratic factor (x ²?px?q) of a polynomial with real coefficients which leaves a remainderr(x), is to determine δp and δq to satisfy

$$0 = rleft( x right) + frac{{partial rleft( x right)}}{{partial P}}delta P + frac{{partial rleft( x right)}}{{partial q}}delta q$$