Please, how do I write this expression : ln(3x + 2) + ln(x + 4) as the logarithm of a single number or expression. This is what I have done:
ln(3x+2) + ln(x + 4)
ln(3x + 2)(x + 4)
3x^2 + 12x + 2x + 8
3x^2 + 14x + 8 = 0
Now I have no clue what to do or even if I am on the right track. Please help me.

So far so good, although you seem to have dropped the "ln" that goes in front, and why did you add "=0" at the end? You're not trying to solve for x here, just put the logarithm expression into a simplified form. In other words, so far you have:

ln(3x^2+14x+8)

Since this is the log of a single expressoin, it seems to me you're done.