You take the same approach as with simplifying other rational expressions like... fractions!
Simplifying 15/6 is a no-brainer for most algebra students.
One method is to break up both the numerator and denominator into factors. --> (3*5)/(2*3)
Obviously, you would cancel out the common factor, 3, giving you 5/2 as the simplest form.
Same principle with ratios of polynomials! You want to create factors and look to cancel out common factors.
15x2 + 24x + 9 can be factored into (3x+3)(5x+3)
**if you need explanation on how to factor this polynomial, see below.
So your factored rational expression now looks like (3x+3)(5x+3)/(3x+3)(1).
Notice that the numerator and denominator both have 3x+3 as a factor, so that factor will cancel out, leaving you with (5x+3)/1 or just 5x+3.
**help on factoring the trinomial in the numerator begins here:
If you have a trinomial that looks like ax2+bx+c, it's called a quadratic trinomial.
I like to call the first and last term, ax2 and c, the "anchors."
To begin, start by thinking of factors for your anchors.
15x2 can be factored into 3x and 5x or x and 15x.
9 can be factored into 3 and 3 or 1 and 9.
So now, you have a few possibilities. A few (not all) of which are:
(3x ± 3)(5x ± 3) or (3x ± 9)(5x ± 1) or (x ± 3)(15x ± 3) or (x ± 1)(15x ± 9)
Now all the above possibilities are based on your "anchors." Here's where your middle term comes in, your bx, or in this case, 24x. When you "FOIL" the above factors, you want the two middle terms to add up to 24x.
As an arbitrary example, say we used (3x+9)(5x+1). If we FOIL, we get 15x2+3x+45x+9 = 15x2+48x+9
Notice, our anchors are "good," but our middle term is not 24x. It's 48x, and that's way off! Now, you could try (3x+9)(5x-1), but that would give you 15x2-3x+45x-9. Now you have the wrong middle term AND the wrong "anchor." (your anchor was a positive 9, not a negative 9, remember?) In order to guarantee a positive 9, you will see that your factors will have to be (a+b)(c+d) or (a-b)(c-d) because (a+b)(c-d) will not give you positive anchors.
It does require some trial-and-error, but with perseverance, you'll eventually find that the correct factors are (3x+3)(5x+3).