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