I need to factor this trinomial

The first step to factoring an expression is to look for a common factor. In this expression we can factor out a from each term to become

a(8a^{2}+14a+3)

Now turn our attention to factoring the expression in the parentheses above

The 8a^{2} term could be factored (8a)(a) or (4a)(2a), the 3 can only be factored as (1)(3).

Which one will give us a 14a as the middle term?

(8a + 1)(a + 3) = 8a^{2} + 25a + 3 Does not work

(8a + 3)(a + 1) = 8a^{2} + 11a + 3 Does not work

(4a + 1)(2a + 3) = 8a^{2} + 14a + 3 WORKS!

So, factored completely, 8a^{3} + 14a^{2} + 3a = **a(4a + 1)(2a + 3)**.