Question

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

Answer 1

Find the roots by calculating the discriminant, you'll find that
`x^2+3x+2 = (x+1)(x+2)`

Explanation:

The discriminant of the quadratic polynomial `p(x)=ax^2+bx+c` is `D=b^2-4ac`

When `D>0` , p(x) has two distinct real roots :
`x1=(-b+sqrtD)/(2a)` and `x2=(-b-sqrt(D))/(2a)`
and `p(x)=(x-x1)(x-x2)`

When `D=0`, p(x) has two coincident real roots
`x1=x2=-b/(2a)`
so `p(x)=(x-x1)^2`

When `D=0`, p(x) has no real roots, but two distinct complex roots
`z{1,2}=\frac{-b \pm i \sqrt {-\D}}{2a}=\frac{-b \pm i \sqrt {4ac-b^2}}{2a}.`

the discriminant of your trinomial `p(x) = x^2+3x+2` is `D=3^2-4*2=1`
`D>0` means you'll have two distinct real roots :
`x = -1 and x=-2`
therefore : `p(x)=(x+1)(x+2)`

Source : https://en.wikipedia.org/wiki/Discriminant