After expanding, line up the $x^3$, $x^2$, $x$, and constant terms on both sides to get equations involving $p$, $q$, $a$, $b$, and $c$.

Pay close attention to the equation that comes from the constant terms (the numbers without $x$). How does that relate $a$ and $c$ to $30$?

From your constant-term equation, think about what must be true about $30$ and $a$ if $c$ is an integer. Which answer choice reflects that relationship?

Create sliders a, b, and c (set them to integer step 1). On a new line, type (x+a)(x^2+bx+c) and then use Desmos’s expand(...) function so you can see the expanded polynomial in standard form.

Look at the expanded result; the constant term will appear as a*c. Since the problem states the constant term is 30, you know a*c = 30. You can also note from the $x^2$ term that p = a + b. Add helper lines like p = a + b and const = a*c if you want.

Add four new expressions: p/a, p/b, 30/a, and 30/b. Move the a, b, and c sliders to different integer values that keep a*c equal to 30 (for instance, try $(a,c) = (2,15)$, $(3,10)$, $(5,6)$, and also negative pairs). Watch which of the four expressions always gives an integer output under this condition; that expression corresponds to the correct answer choice.

Start with the given factored form:

Multiply term by term:

We are told this equals the polynomial

Matching coefficients of corresponding powers of $x$ gives:

• Coefficient of $x^2$: $p = a + b$.
• Coefficient of $x$: $q = ab + c$.
• Constant term: $30 = ac$.

We also know from the problem that $a$, $b$, and $c$ are integers.

From the constant terms we have

Since $a$ and $c$ are both integers and their product is $30$, we can solve for $c$:

Because $c$ is given to be an integer, $\dfrac{30}{a}$ must always be an integer. The other choices involve divisions like $\dfrac{p}{a}$ or $\dfrac{30}{b}$, which are not guaranteed to be integers for all integer values of $a$, $b$, and $c$ that satisfy the conditions. Therefore, the expression that must be an integer is $\dfrac{30}{a}$ (choice C).