If (x + 1) (y + 1) (y + x) + xy = 0

x + y ≠ -xy

Find the value of (x + y + 4)²  

9

(x + 1) (y + 1) (y + x) + xy = 0  ..... (i)

if x + y ≠ -xy

it means (x + y + xy) is one of the root of the given equation (i)

So,

(x + 1) (y + 1) (y + x) + xy = 0

(xy + y + x + 1) (y + x) + xy = 0

xy (y + x) + (y + x + 1)(y + x) + xy = 0

xy (y + x + 1) + (y + x + 1)(y + x) = 0

(y + x + 1)(xy + y + x) = 0

(x + y + xy) (x + y + 1) = 0

So, x + y + 1 = 0 and x + y = -1

Put in (x + y + 4)²  

⇒ (-1 + 4)² = 9