Factorize x2+4xy+4y2-9x2y2
if x = 2, y = -1, then the value of x^2 + 4xy + 4y^2 is
x^2 + 4xy + 4y^2 - 9x^2y^2
= (x+2y)^2 - 9x^2 y^2
= (x+2y)^2 - (3xy)^2
now you have a difference of squares
= (x+2 + 3xy)(x+2 - 3xy)
Good
To factorize the expression x^2 + 4xy + 4y^2 - 9x^2y^2, we can first group the terms and look for common factors.
Notice that the first three terms, x^2, 4xy, and 4y^2, have a common factor of (x + 2y). We can factor out this common factor by applying the distributive property:
(x + 2y)(x + 2y) + 4y^2 - 9x^2y^2
Now, we have a quadratic expression (x + 2y)(x + 2y), which can further be simplified by recognizing that it is a perfect square. We can rewrite it as:
(x + 2y)^2 + 4y^2 - 9x^2y^2
To proceed, let's focus on the last two terms, 4y^2 and -9x^2y^2. Notice that both terms contain a common factor of y^2. We can factor out this common factor:
(x + 2y)^2 + y^2(4 - 9x^2)
At this point, we have factored out the common factors. However, we can simplify the expression further by looking at the term (4 - 9x^2). It does not have any common factors, but we can recognize that it is a difference of squares. We can rewrite this term as:
(x + 2y)^2 - (3xy)^2
Now, we have a difference of squares expression. We can use the formula a^2 - b^2 = (a + b)(a - b) to factorize it:
[(x + 2y) + 3xy][(x + 2y) - 3xy]
Therefore, the factored form of the expression x^2 + 4xy + 4y^2 - 9x^2y^2 is:
[(x + 2y) + 3xy][(x + 2y) - 3xy]