After you clear the denominator and expand $(x-2)(x^2 + px + q)$, line up the resulting polynomial with $x^3 + ax^2 + bx + c$. How can you express $a$, $b$, and $c$ in terms of $p$ and $q$?

Translate "consecutive integers in increasing order" into equations that relate $a$, $b$, and $c$. Then combine these with your earlier expressions for $a$, $b$, and $c$ in terms of $p$ and $q$ to get equations involving just $p$ and $q$.

You should end up with two equations that link $p$ and $q$. Solve this system step by step—solve one equation for one variable, substitute into the other, and then find $q$.

After you have found specific integer values for $a$, $b$, $c$, $p$, and $q$, type the left side as (x^3 + a*x^2 + b*x + c)/(x - 2) and the right side as x^2 + p*x + q + 5/(x - 2) in Desmos. If your work is correct, the two graphs should overlap completely (they represent the same function for all $x \neq 2$); if they do not, recheck your algebra and the value of $q$.

Because the equation holds for all real $x \neq 2$, you can multiply both sides by $x-2$:

becomes

Now both sides are polynomials in $x$, equal for all $x$, so their coefficients must match.

First expand $(x-2)(x^2 + px + q)$:

Then add the $+5$:

So

Matching coefficients of like powers of $x$ gives the system

• $a = p - 2$
• $b = q - 2p$
• $c = -2q + 5$.

The problem says $a,b,c$ are consecutive integers in increasing order, so

• $b = a + 1$
• $c = a + 2$.

Use $a = p - 2$ from the previous step:

• $b = a + 1 = (p - 2) + 1 = p - 1$
• $c = a + 2 = (p - 2) + 2 = p$.

But from coefficient matching you also have

• $b = q - 2p$
• $c = -2q + 5$.

So you get two equations relating $p$ and $q$:

• $q - 2p = p - 1$
• $-2q + 5 = p$.

These two equations are enough to solve for $p$ and then $q$.

Solve $q - 2p = p - 1$:

Substitute this into $-2q + 5 = p$:

Now plug $p = 1$ back into $q = 3p - 1$:

So the value of $q$ is $\boxed{2}$.