Answer
$(x^4+1)(x+2)(x-2)(x^2+4)$
Work Step by Step
The product of a binomial sum and a binomial difference is: $(A+B)(A-B)=A^2-B^2$.
Hence here: $x^8-15x^4-16=\\=x^8+x^4-16x^4-16\\=x^4(x^4+1)-16(x^4+1)\\=(x^4+1)(x^4-16)\\=(x^4+1)((x^2)^2-4^2)\\=(x^4+1)(x^2-4)(x^2+4)\\=(x^4+1)(x^2-2^2)(x^2+4)\\=(x^4+1)(x+2)(x-2)(x^2+4)$
