Question

How do you factor `8x^3y^6 + 27`?

Answer 1

`a^3+b^3 = (a+b)(a^2-ab+b^2)`
(this is either something you already know or you can derive it by synthetic division/multiplication)

`8x^3y^6+27 = (2xy^2)^3+3^3`

So this can be factored as
`(2xy^2+3)((2xy^2)^2-(2xy^2)(3)+3^2)`
or
`(2xy^2+3)(4x^2y^4-6xy^2+9)`

You might be tempted to try to factor the second part of this but if you consider the formula for roots of a quadratic:
`(-b+-sqrt(b^2-4ac))/(2a)`

we have the square root of a negative value; so there are no Real factors available.