When the coefficient of the x squared term is not 1 you need to use the method of decomposition to factor the trinomial.
The first part you have done correctly.
Find two numbers whose product is a*c and whose sum is b.
In this case a*c=2 and b=3 so the two numbers are 2 and 1.
Now, since the x squared coefficient is not 1 you must use the method of decomposition to factor the trinomial.
You replace the middle term with the two numbers you found. So 2a^2+3a+1 gets rewritten as 2a^2 +2a + a + 1.
Then group the first two terms and last two terms factoring out the largest factor you can out of each group.
So you have 2a(a+1)+(a+1).
Now take (a+1) out as a common factor giving (a+1)(2a+1)
Here is a link with a very clear example of how this works.
http://mathcentral.uregina.ca/QQ/databa ... /kim1.html