Answer:
(x + 5 y)^2
Step-by-step explanation:
Simplify the following:
4 (x - y)^2 - 12 (x - y) (x + y) + 9 (x + y)^2
(x - y) (x - y) = (x) (x) + (x) (-y) + (-y) (x) + (-y) (-y) = x^2 - x y - x y + y^2 = x^2 - 2 x y + y^2:
4 x^2 - 2 x y + y^2 - 12 (x - y) (x + y) + 9 (x + y)^2
4 (x^2 - 2 x y + y^2) = 4 x^2 - 8 x y + 4 y^2:
4 x^2 - 8 x y + 4 y^2 - 12 (x - y) (x + y) + 9 (x + y)^2
(x + y) (x - y) = (x) (x) + (x) (-y) + (y) (x) + (y) (-y) = x^2 - x y + x y - y^2 = x^2 - y^2:
4 x^2 - 8 x y + 4 y^2 - 12 x^2 - y^2 + 9 (x + y)^2
-12 (x^2 - y^2) = 12 y^2 - 12 x^2:
4 x^2 - 8 x y + 4 y^2 + 12 y^2 - 12 x^2 + 9 (x + y)^2
(x + y) (x + y) = (x) (x) + (x) (y) + (y) (x) + (y) (y) = x^2 + x y + x y + y^2 = x^2 + 2 x y + y^2:
4 x^2 - 8 x y + 4 y^2 - 12 x^2 + 12 y^2 + 9 x^2 + 2 x y + y^2
Grouping like terms, 4 x^2 - 8 x y + 4 y^2 - 12 x^2 + 12 y^2 + 9 (x^2 + 2 x y + y^2) = 9 (x^2 + 2 x y + y^2) + (4 y^2 + 12 y^2) - 8 x y + (4 x^2 - 12 x^2):
9 (x^2 + 2 x y + y^2) + (4 y^2 + 12 y^2) - 8 x y + (4 x^2 - 12 x^2)
4 y^2 + 12 y^2 = 16 y^2:
9 (x^2 + 2 x y + y^2) + 16 y^2 - 8 x y + (4 x^2 - 12 x^2)
4 x^2 - 12 x^2 = -8 x^2:
9 (x^2 + 2 x y + y^2) + 16 y^2 - 8 x y + -8 x^2
9 (x^2 + 2 x y + y^2) = 9 x^2 + 18 x y + 9 y^2:
9 x^2 + 18 x y + 9 y^2 + 16 y^2 - 8 x y - 8 x^2
Grouping like terms, 9 x^2 + 18 x y + 9 y^2 + 16 y^2 - 8 x y - 8 x^2 = (9 y^2 + 16 y^2) + (18 x y - 8 x y) + (9 x^2 - 8 x^2):
(9 y^2 + 16 y^2) + (18 x y - 8 x y) + (9 x^2 - 8 x^2)
9 y^2 + 16 y^2 = 25 y^2:
25 y^2 + (18 x y - 8 x y) + (9 x^2 - 8 x^2)
x y 18 + x y (-8) = 10 x y:
25 y^2 + 10 x y + (9 x^2 - 8 x^2)
9 x^2 - 8 x^2 = x^2:
25 y^2 + 10 x y + x^2
The factors of 25 that sum to 10 are 5 and 5. So, 25 y^2 + 10 x y + x^2 = (x + 5 y) (x + 5 y):
(x + 5 y) (x + 5 y)
(x + 5 y) (x + 5 y) = (x + 5 y)^2:
Answer: (x + 5 y)^2
