Question

How do you factor `x^2 + 2x +3`?

Answer 1

There are no integer factors for this expression.

Explanation:

There are no integers which are factors.

We would need factors of 3 which add up to 2. There are simply none.

Answer 2

`x^2+2x+3=(x+1+isqrt2)(x+1-isqrt2)`

Explanation:

Zeros of `ax^2+bx+c` are given by quadratic formula `(-b+-sqrt(b^2-4ac))/(2a)`, however, such a quadratic function can be factorized, if the discriminant `(b^2-4ac)` is square of a rational number.

In `x^2+2x+3`, discriminant is `2^2-4*1*3=4-12=-8` and hence negative. So its zeros are two complex conjugate numbers given by quadratic formula i.e.

`(-2+-sqrt(2^2-4*1*3))/2` or

`(-2+-sqrt(-8))/2` or

`-1+-isqrt2` i.e. `-1-isqrt2` and `-1+isqrt2`

Now, if `alpha` and `beta` are zeros of quadratic polynomial, then its factors are `(x-alpha)(x-beta)`

Hence factors of `x^2+2x+3` are `(x+1+isqrt2)` and `(x+1-isqrt2)` and

`x^2+2x+3=(x+1+isqrt2)(x+1-isqrt2)`