Answer
$(x-1)(x^2+x+1)(x-2)(x^2+2x+4)$
Work Step by Step
The formula for factoring the sum of two cubes is: $A^3+B^3=(A+B)(A^2-AB+B^2)$.
The formula for factoring the difference of two cubes is: $A^3-B^3=(A-B)(A^2+AB+B^2)$.
Hence here: $x^6-9x^3+8=\\=x^6-x^3-8x^3+8\\=x^3(x^3-1)-8(x^3-1)\\=(x^3-1)(x^3-8)\\=(x^3-1^3)(x^3-2^3)\\=(x-1)(x^2+x+1)(x-2)(x^2+2x+4)$