After you factor out the common $(x+2)$ in the numerator, what happens when you divide by $(x+2)$ in the denominator?

When you subtract $(13x^2 - 9x + 30)$ from $(13x^2 + 11x - 24)$, remember that the minus sign distributes to every term in the second parentheses.

Once you have a linear expression of the form $ax + b$, match $a$ with $p$ and $b$ with $q$, then compute $p - q$.

Type the full expression into Desmos as a function, for example:

Then create a table for $f(x)$ (click the gear icon and "Convert to table") to see values such as $f(0)$ and $f(1)$.

Since the simplified form is $px+q$, note that $f(0) = q$ and $f(1) = p + q$. From your table values, compute $p = f(1) - f(0)$ and then $p - q = [f(1) - f(0)] - f(0) = f(1) - 2f(0)$. Use your Desmos values for $f(0)$ and $f(1)$ to get the numerical value of $p - q$.

Look at the numerator:

Both terms share a common factor of $(x+2)$, so factor it out:

Now the whole expression becomes

For $x \ne -2$, the $(x+2)$ in the numerator and denominator cancel, leaving just the bracketed expression.

After canceling $(x+2)$, you have

Distribute the minus sign over the second parentheses:

Now combine like terms:

• $13x^2 - 13x^2 = 0$ (the $x^2$ terms cancel),
• $11x + 9x = 20x$,
• $-24 - 30 = -54$. So the simplified expression is

This matches the form $px+q$ with specific values of $p$ and $q$.

From the simplified expression $20x - 54$:

• $p$ is the coefficient of $x$, so $p = 20$,
• $q$ is the constant term, so $q = -54$. Now compute

So, the value of $p - q$ is 74.