Factor the expression over the complex numbers. x^2+36

Note that

a² - b² = (a+b) (a-b)

i² = - 1 by definition.

We want to factorize x² + 36 = x² + 6².

Consider the expression

x² - (6i) ² = x² - (6²) (i²) = x² - 36i² = x² + 36

Because x² - (6i) ² = (x + 6i) (x - 6i)

Therefore

x² + 36 = (x + 6i) (x - 6i)

Answer: (x + 6i) (x - 6i)