Question

How do you factor the expression `16x² + 48xy +36y²`?

Answer 1

`16x^2+48xy+36y^2 = 4(2x+3)^2`

Explanation:

First note that all of the terms are divisible by `4`, so separate that out as a factor:

`16x^2+48xy+36y^2 = 4(4x^2+12xy+9y^2)`

Next note that both `4x^2 = (2x)^2` and `9y^2 = (3y)^2` are perfect squares. So if we square `(2x+3)` then the resulting first term will be `4x^2` and the last term `9y^2`. How about the middle term?

`(2x+3)^2 = (2x)^2+2(2x)(3y)+(3y)^2`

`color(white)((2x+3)^2) = 4x^2+12xy+9y^2`

So we have a perfect square trinomial.

Putting it all together:

`16x^2+48xy+36y^2 = 4(2x+3)^2`