Calculus with Applications (10th Edition)

$\frac{3(x-1)}{x-2}$
Step 1: The first thing you have to do to simplify this fraction is to factor the numerator and the denominator. To factor $3x^{2}+3x-6$, you first want to factor out the greatest common factor (the highest value that is found in all of the terms) which in this case is 3 (since 3 is the only thing that is in all of the terms). Once you factor out the 3, you are left with: $x^{2}+x-2$. However, you can still factor that as well. To factor a trinomial, you want to first look at the first term which is $x^{2}$. Since the coefficient of $x^{2}$ is 1, you factor by finding two numbers whose product is the 3rd term (so in this case -2) and whose sum is the middle term (so in this case 1). The only possible pair of numbers that multiply together to get -2 but add to get a positive 1 is -1 and 2. Therefore, $x^{2}+x-2$ factors as: (x-1)(x+2). Next, you have to factor the denominator which is $x^{2}-4$. Since both $x^{2}$ and 4 are perfect squares and they are being subtracted, you can factor by using the rules for difference of squares. To factor, you take the square of each separate term and put it into two separate brackets - one addition and one subtraction. As a result, $x^{2}-4$ fully factored is (x-2)(x+2). Step 2: Now that the numerator and denominator are factored, you are left with $\frac{3(x+2)(x-1)}{(x-2)(x+2)}$. You can then cancel out like terms in the numerator and denominator. Since (x+2) is in both, you can cancel it out because anything divided by itself equals 1. Since there is no more common terms, your final answer is $\frac{3(x-1)}{x-2}$.