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Well lets try foiling your given answer and see that you get.

1) 3a(2a - 3) (a + 4)

2) 3a(2a*a - 3a + 8a - 12)

3) 3a(2a^2 + 5a - 12)

So this solution is correct. So what happened? I want you to pay attention to 2a in equation #1, notice how by foiling the outside numbers, we got 2a*4= 8a. This is the reason that the factors that you are given worked. So how do we figure out this factor from looking at the equation

3a(2a^2 + 5a - 12)

I'm gonna ignore the 3a because you understand that part and just focus on factoring the parentheses.

(2a^2 + 5a - 12) = ( ___ + ___ )( ___ + ___ )

So lets do this one at a time. On the red blanks, we have to put two numbers that multiply together to get 2a^2, well since the factors of 2 are 2*1 there is only one way you can do this. Realize that 2a^2=2a*1a and that is what has to go on our red blanks

(2a^2 + 5a - 12) = ( 2a+ ___ )( a + ___ )

ok now the second 2 blanks are a little bit tricky you have to put 2 numbers that multiply together to get -12, so the first thing to notice is it has to be a negative x a positive to get a negative. Ok so what are the factors of 12? -12= -1*12, -2*6 and -3*4 (note you can switch around the negatives as needed). Now here your correct, none of these add up to 5, but that's because we haven't taken in to account that sneaky 2a yet.

So what does that 2a mean?
It means instead of checking if any of these factors add up to 5
-1+12 = 11
-2+6 = 4
-3+4 = 1

You have to multiply one of the factors by 2 to account for that extra 2a. Now the question is which one?
2(-1)+12 = 10
-1+2(12) = 23
2(-2)+6 = 2
-2+2(6) = 10
2(-3)+4 = -2
-3+2(4) = 5 <- aha, so we found the correct factors, so choose -3 and 4 to go in our green spaces.

(2a^2 + 5a - 12) = ( 2a+ (-3) )( a + 4 )

ok so this method is a little labor intensive, but there's a quicker method. A trick that you can use.

take the first number and the last number (2a^2 and -12), ignore the a^2 for a second just looking at 2 and -12. (BTW the 2 on the 2a^2 is called the "coefficient" and so is the 5 on the 5a)
Multiply 2 and -12 together. 2*-12= -24, now we want to find factors of -24 that add up to 5. -24= -1*24, -2*12, -3*8, -4*6

-1+24 = 23
-2+12= 10
-3+8 = 5 <- We're done now, so we don't have to check the last one. (note that if we had gotten a -5 here, ie -8+3=-5, all we have to do is switch the negatives to get the correct factors

now -3+8 is not quite our answer, we just have to remember again that pesky 2a. All we have to do is factor out a two out of one of these factors, since 8 is the only one of them that we can factor out a 2, we can rewrite -3+8 as -3+2(4)
so -3 and 4 are our factors (which multiplies to get -12 as desired) and the multiplication by 2 accounts for our 2a. So our answer again is

(2a-3)(a+4)

(this is referred to as the AC method since an arbitrary polynomial is written as Ax2+Bx+C. Where in this case, 2 was our A and -12 was our C (also 5 was B). And the method is to multiply A x C which is just AC.)