Rewrite $4x^2-9$ in the form $a^2-b^2$. What are $a$ and $b$, and what does $a^2-b^2$ factor into?

After factoring the numerator, see if there is a common factor with the denominator $2x+3$ that you can cancel (remember $x \ne -\tfrac{3}{2}$). What simple expression is left?

Once you have a simplified linear expression, compare it to $px+q$. The number multiplying $x$ is $p$, and the constant term is $q$. Identify $q$.

Type y = (4x^2 - 9)/(2x + 3) into Desmos. This will show the graph of the rational expression (with a hole where the denominator is zero).

Notice that for large $x$, the graph looks like a straight line with slope $2$, so the equivalent linear expression has the form $2x + q$. Now, in new lines, graph each candidate line: y = 2x - 6, y = 2x - 3, y = 2x + 3, and y = 2x + 6. The correct line is the one that perfectly overlaps the rational graph everywhere it is defined (except at the hole). Read off the constant term $q$ from that matching line.

We are told that

can be written in the form $px+q$, where $p$ and $q$ are constants. That means we need to simplify the rational expression until it becomes a linear expression (something like $ax+b$).

Notice that $4x^2-9$ is a difference of squares:

• $4x^2 = (2x)^2$
• $9 = 3^2$ So we can factor the numerator as

Now the expression becomes

The factor $2x+3$ appears in both the numerator and the denominator. For $x \ne -\tfrac{3}{2}$ (so that $2x+3 \ne 0$), we can cancel this common factor:

So the original expression is equivalent to the linear expression $2x-3$ (except at $x=-\tfrac{3}{2}$, where the original is undefined).

We now have the simplified form

Comparing this with $px+q$:

• The coefficient of $x$ is $p=2$.
• The constant term is $q=-3$.

Therefore, the value of $q$ is $-3$, which corresponds to choice B.