Factor the trinomial step by step & explain 2x^2 + 5x + 3

One way to factor equation is to find the zeros. Its obvious that x=-1 is solution for this. So one factor is (x+1)

the next factor should include 2x at first (because we have 2x^2 in the equation which can not be made any other way)

Let's suppose the factor is (x+1) (2x+b)

Since we do not know what is b open the brackets 2x^2+bx+2x+b. If this is equal to 2x^2+5x+3 then b=3. We are left with (x+1) (2x+3) = 2 (x+1) (x+3/2)

Would go for other shorter solutions, but they require some deeper understanding like Horner's scheme, Fermat's Theorem or even deeper which I assume You will not understand)