Notice that the denominator $q(x)$ is $x^2-4$. Can you rewrite the numerator $p(x)$ so that $x^2-4$ appears as a factor in it?

Try writing $p(x)$ as $(x^2-4)(ax^2+bx+c)$. Expand this product and compare its coefficients with those of $p(x)$ to find $a$, $b$, and $c$.

In Desmos, enter

Optionally, you can add {x>2} at the end to focus on the domain in the problem: y1 = (x^4+5x^3-10x^2-20x+24)/(x^2-4) {x>2}.

Enter each choice as its own function (again you can restrict to $x>2$ if you like), for example:

• $y_2 = x^2+5x+6$
• $y_3 = x^2+5x-\dfrac{6}{x^2-4}$
• $y_4 = x^2+5x-6$
• $y_5 = \dfrac{x^2+5x-6}{x+2}$

Adjust the window so you can clearly see the graphs for $x>2$.

Look at the curves for $x>2$ and see which answer choice’s graph lies exactly on top of the graph of $y_1$ (the original expression) for all visible $x$ in that range. The choice whose graph perfectly overlaps $y_1$ on $x>2$ is the equivalent expression.

We want an expression that is equivalent to

for $x>2$.

Instead of plugging in numbers, the most direct way is to simplify this rational expression by either

• factoring the numerator so it includes $x^2-4$ as a factor, or
• performing polynomial long division by $x^2-4$.

We will use factoring with unknown coefficients.

Suppose $p(x)$ can be written as $(x^2-4)$ times some quadratic:

for some constants $a, b, c$. If this is true, then

whenever $x^2-4 \ne 0$.

Next we will expand the right-hand side and match coefficients with $p(x)$.

First expand the product:

We want this to equal

So we match coefficients of each power of $x$:

• $x^4$ term: $a = 1$.
• $x^3$ term: $b = 5$.
• $x$ term: $-4b = -20$, which confirms $b = 5$.
• Constant term: $-4c = 24$, so $c = -6$.
• Check $x^2$ term: $c - 4a = -6 - 4(1) = -10$, matching the $-10x^2$ in $p(x)$.

Now we know the exact quadratic $ax^2+bx+c$.

Using $a=1$, $b=5$, and $c=-6$, we have

Therefore,

for all $x$ such that $x^2-4 \ne 0$ (that is, $x \ne \pm 2$).

Since the problem restricts to $x>2$, the denominator is never zero, so the expression is equivalent to $x^{2}+5x-6$ on the given domain. This corresponds to answer choice C.