Question

Factors of x 2 + 4y 2 +4y - 4xy - 2x -8

Answer 1

Let the given polynomial be the product of :
    (x + b y + c) ( x + 4/b y - 8/c) Expand that:
      x^2 + (4/b + b) x y + (c - 8 /c) x + 4 y^2 + (4c/b - 8 b/c) y - 8compare coefficients and equate them:

          4 + b^2  + 4 b= 0   .... 1
          c^2 - 8  + 2 c = 0    .....2
          c^2 - 2 b^2 - bc = 0   ..... 3

Solving them , easily: b = - 2     c = 2  or - 4   
Factors are :
x^2 + 4 y^2 + 4 y - 4 x y - 2 x - 8
  =  (x -2 y + 2 ) (x -2 y - 4)   OR   (x -2 y - 4) (x -2 y + 2 )