Question

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

Answer 1

Since `2x^2+5x+3` is of the form `ax^2+bx+c`
we could use the formula
`(-b+-sqrt(b^2-4ac))/(2a)`

But if our answer only involves integers (no guarantee of this!) there may be a simpler method)

Suppose the factors look like
`(2x+p)(x+q)`
The coefficient of `x` will be
`2q+p = 5` (which implies both `p` and `q` can't be negative)
and
the term which does not include `x` will be
`pq = 3` which implies `p` and `q` must have the same sign.
(therefore `p` and `q` must both be positive).

If the solution values are restricted to integers there are only two possibilities given that `p` and `q` are both `>0` and that `pq=3`

`(p,q) = (1,3)` or `(p,q)=(3,1)`

If `(p,q) = (1,3)`
then `2q+p = 2(3)+1 != 5`
so this is incorrect.

If `(p,q) = (3,1)`
then `2q + p = 2(1) + 3 = 5`
and these are the required values.

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