%5E2-4(x-y)(x%2B2y)%2B4(x%2B2y)%5E2.jpeg "(x-y)^2-4(x-y)(x+2y)+4(x+2y)^2")
Factoring the Expression: (x-y)^2 - 4(x-y)(x+2y) + 4(x+2y)^2
This expression might look complicated at first glance, but it can be factored quite easily by recognizing a pattern and applying some algebraic manipulations.
Identifying the Pattern
Notice that the expression resembles a perfect square trinomial. This pattern arises when squaring a binomial:
(a + b)^2 = a^2 + 2ab + b^2 (a - b)^2 = a^2 - 2ab + b^2
Let's rearrange our expression to make the pattern more apparent:
(x-y)^2 - 4(x-y)(x+2y) + 4(x+2y)^2 = (x-y)^2 - 2(2)(x-y)(x+2y) + (2(x+2y))^2
Applying the Pattern
Now we can see the pattern clearly. We have:
- a = (x - y)
- b = 2(x + 2y)
Using the perfect square trinomial formula, we can factor the expression as:
(a - b)^2 = [(x-y) - 2(x+2y)]^2
Simplifying the Expression
Let's simplify the expression further:
[(x-y) - 2(x+2y)]^2 = (-x - 5y)^2 = (x + 5y)^2
Therefore, the factored form of the expression (x-y)^2 - 4(x-y)(x+2y) + 4(x+2y)^2 is (x + 5y)^2.