Question

How do you factor `243(3x - 1)^2 - 48(2y + 3)^2`?

Answer 1

Use the difference of squares property to get `3(27x+8y+3)(27x-8y-21)`.

Explanation:

What should always jump out at you in a factoring question containing a minus sign and stuff squared is difference of squares:
`a^2-b^2=(a-b)(a+b)`

But the 243 and 48 kind of kill that idea, because they aren't perfect squares. However, if we factor out a `3`, we have:
`3(81(3x-1)^2-16(2y+3)^2)`

Which can be rewritten as:
`3((9(3x-1))^2-(4(2y+3))^2)`

Now we can apply difference of squares, with:
`a=9(3x-1)`
`b=4(2y+3)`

Doing so gives:
`3((9(3x-1))^2-(4(2y+3))^2)`
`=3((9(3x-1)+4(2y+3))(9(3x-1)-4(2y+3))`

Let's get rid of some parentheses by distributing:
`3((9(3x-1)+4(2y+3))(9(3x-1)-4(2y+3))`
`=3(27x-9+8y+12)(27x-9-8y-12)`

Finally, collect terms:
`3(27x-9+8y+12)(27x-9-8y-12)`
`=3(27x+8y+3)(27x-8y-21)`