Question

How do you factor `(x^2+5)(x^2+2x-3)`?

Answer 1

`(x^2+5)(x^2+2x-3) = (x^2+5)(x+3)(x-1)`

The factors `(x+3)` and `(x-1)` were found by looking for two numbers `a` and `b` such that `axxb = 3` and `a-b = 2`.
Then `(x+a)(x-b) = x^2+(a-b)x-ab`.

This is as far as you can break down the original expression into factors, unless you are allowed complex numbers as coefficients. You can tell that `(x^2+5)` has no smaller factors with real coefficients - which would be linear - because `x^2+5 > 0` for all real values of `x`.

On the other hand, if you are allowed complex coefficients, you can factor as:

`(x^2+5) = (x+isqrt(5))(x-isqrt(5))`, where `i = sqrt(-1)`, so

`(x^2+5)(x^2+2x-3)`

`= (x+isqrt(5))(x-isqrt(5))(x+3)(x-1)`