Factoring

$3x^2+5x+2\\ $

First you need to find two numbers that multiply to give 3*2=6  

Obviously they need to be of the same sign (positive or negative)

They also need to add to 5.    Since 5 is positive both numbers must be positive.

2 and 3 work

2+3=5

2*3=6

so now you need to write 5x as 3x+2x

$3x^2+5x+2\\ =3x^2+3x+2x+2\\ =3x^2+3x\quad+\quad2x+2\\ \text{Now factor in pairs}\\ =3x(x+1)\quad + \quad 2(x+1)\\ =(3x+2)(x+1)$