and

and

and

and

The expression is a perfect square trinomial, as the three terms have the following relationship:

We can factor this expression by substituting  into the following pattern:

We can factor further by noting that , the difference of squares, and subsequently, factoring this as the product of a sum and a difference.

or 

Note that 

Therefore  is equivalent to:

The correct answer is . Our work proceeds as follows:

   (Factor an  out of the numerator and denominator)

 (Factor the quadratic polynomials)

 (Cancel common terms)

 (Multiply by  to both sides)

   (Distribute the )

 (Simplify and solve)

In order to solve for  we must first factor: 

The zero product property states that if  then  or  (or both).

Our two equations are then:

Solving for  in each leaves us with:

and 

The roots of a function are the points at which it crosses the x axis, so at these points the value of y, or f(x), is 0. This gives us:

So we will have to factor the polynomial in order to solve for the x values at which the function is equal to 0. We need two factors whose product is -18 and whose sum is -3. If we think about our options, 2 and 9 have a product of -18 if one is negative, but there's no way of making these two numbers add up to -3. Next we consider 3 and 6. These numbers have a product of -18 if one is negative, and their sum can also be -3 if the 3 is positive and the 6 is negative. This allows us to write out the following factorization:

We could solve this question a variety of ways. The simplest would be graphing with a calculator, but we will use factoring. 

To begin, set our function equal to . We want to find where this function crosses the -axis—in other words, where .

Next, we need to factor the function into two binomial terms. Remember FOIL/box method? We are essentially doing the reverse here. We are looking for something in the form of .

Recalling a few details will make this easier.

1)  must equal positive 

2)  and  must both be negative, because we get positive  when we multiply them and  when we add them.

3)  and  must be factors of  that add up to . List factors of : . The only pair of those that will add up to  are  and , so our factored form looks like this:

Then, due to the zero product property, we know that if  or  one side of the equation will equal , and therefore our answers are positive  and positive .

If you note that the expressions on the left sides of the equations are perfect square trinomials, you can rewrite the expressions as follows:

Either  or 

Similarly,

Either  or 

Four systems of equations can be set up here, and in each case, we can find  by adding both sides:

The set of possible values of  is .