How to algebraically factor a polynomial, for example 3x²+7x+2

Try to get it into form

(ax+b) (cx+d)

a, b, c, d can be negative or positive

3x^2+7x+2

2 and 7 are positive so b and d will be positive as well

3 is prime so we have to have

(3x+?) (1x+?)

2 is also prime

try

(3x+2) (1x+1), expanded it is 3x^2+5x+2, nope

(3x+1) (x+2), expanded is 3x^2+7x+2, yep

mostly a trial and error sort of thing

after some practice, it might get easier for you to see