Factoring using Difference of cubes

Q:  How do I factor this:  27 – 64x⁶

A:  This looks like a difference of cubes… How do I know?  It looks like (something cubed) – (something else cubed).

The difference of cubes formula is one you have to memorize or just have handy… It says:

a³ – b³ = (a – b)(a² + ab + b²)

[I color-coded it to make it easier for the next steps].

OK, back to our original problem…  I need to make it look like (something)³ – (something)³

27 – 64x⁶ = 3³ – (4x²)³

So, we have:

3³ – (4x²)³ = (3 – 4x²)(3² + 3*4x² + (4x²)²)

Now, simplify the right side:

(3 – 4x²)(3² + 3*4x² + (4x²)²) = (3 – 4x²)(9 + 12x² + 16x⁴)

There it is!