How to factor this?
Most people trying to factor this expression would say that it is irreducible, that is, it cannot factor at all. I will show you a trick that consist of including two terms that not appear in the original expression.
x4 + 64 = x4 + 16x2 - 16x2 + 64
The two terms 16x2 and -16x2 appear by considering the square root of x4 and by dividing the original constant by the leading exponent (64 ÷ 4 = 16). Then, you change the position of the negative term to the last position:
x4 + 16x2 - 16x2 + 64 = x4 + 16x2 + 64 - 16x2
The first three terms form a perfect square trinomial, which can be easily factored. The last term can be rewritten considering the square root concept.
x4 + 16x2 + 64 - 16x2
= (x2 + 8)(x2 + 8) - (4x)2
= (x2 + 8)2 - (4x)2
The preceding expression is a difference of two sqaures. Considering the pattern for this type of factorizaion (a2 - b2), a equals x2 + 8 while b = 4x. So, knowing that a2 - b2 = (a + b)(a - b), we have
(x2 + 8)2 - (4x)2 = (x2 + 8 + 4x)(x2 + 8 - 4x),
which gives the solution to the exercise.
I'll hope that this helping tool would lead you to solve any similar type of factorization exercises.