Question

How do you factor `x^2 + 4x + 3`?

Answer 1

The factors are `(x+3)` and `(x+1)`

Explanation:

We are asked for factorise: `x^2+4x+3`

First notice that the function is a quadratic and so will have two factors. Since the coefficient of `x^2` is 1, the factors will be of the form: `(x+a)(x+b)` We will assume that a and b are integers.

Hence, we need to find a and b such that the product of the factors is equal to the given quadratic function .

Now consder the absoute value of constant term, 3. Since 3 is prime its only factors are 3 and 1. Since the constant term is positive, a and b can only be 3 and 1 or -3 and -1.

Finally observe that the coefficient of `x` is positive 4 and that the sum of 3 and 1 is positive 4. Thus, a and b must be 3 and 1 (or the other way around, but this makes no difference to our factorisation)

Hence we see that: `x^2+4x+3` = `(x+3)(x+1)`

Therefore the factors are `(x+3)` and `(x+1)`

Answer 2

(x + 1)(x + 3)

Explanation:

y = x^2 + 4x + 3
Since a - b + c = 0, use shortcut:
- One real root is (-1) and the factor is (x + 1)
- The other real root is (-c/a = - 3), and the factor is (x + 3)
Therefor,
y = (x + 1)(x + 3)