Question

How do you factor 3x^3+81

Answer 1

Call it 3(x^3 + 27).

27 is the cube of three, so one root is x = -3 and the corresponding factor is (x+3). For the other factor, use syntheic division to compute
(x^3+27)/(x+3). That turns out to be
(x^2 -3x + 9). The other roots are complex, so leave it as

3x^3 + 81 = 3(x+3)(x^2 -3x + 9)