Question

How do you factor completely `48x^2 + 48xy^2 + 12y^4`?

Answer 1

`12(2x+y^2)^2`

Explanation:

Step one: Factor out the greatest common factor (which is 12):
`12(4x^2+4xy^2 + y^4)`

Step two: Recognize that this is a perfect square. Remember that Perfect Square Trinomials have:
a) first and last terms that are positive perfect squares (in this case, `4x^2` and `y^4`.
AND
b) the middle term is twice the product of the square roots of the first and third terms (in this case `2xy^2 xx 2 = 4xy^2`

When you have a perfect square, you can factor it by taking the square root of the first and last terms, putting them inside a bracket, and squaring the bracket:

`(2x+y^2)^2`

Step three: Putting the 12 in front (that you factored out in the first step) completes the answer:
`12(2x+y^2)^2`