Think of $(a+b)(c+d)$: every term in the first parentheses must multiply every term in the second. Do the same with $\frac{3}{4}x - \frac{5}{6}$ and $8x + 9$.

You will get two $x$ terms with fractional coefficients. Rewrite them with a common denominator before adding, and keep track of the minus sign on $-\frac{5}{6}$ when multiplying.

After simplifying, make sure your final expression has the correct $x^2$ term, the correctly combined $x$ term, and the correct constant term before choosing an option.

In Desmos, type p(x) = (3/4)x - 5/6 and q(x) = 8x + 9 on two separate lines so that $p(x)$ and $q(x)$ are defined.

On a new line, type r(x) = p(x)*q(x) so Desmos defines the product function $r(x) = p(x)\cdot q(x)$.

Pick a value for $x$, for example $x = 1$. Type r(1) to find the value of the product at $x=1$. Then, for a choice (say choice A), define it as a function, for example A(x) = 6x^2 + (1/12)x - 15/2, and type A(1). If $A(1)$ equals $r(1)$, that choice is a candidate.

To be confident, test another value such as $x=2$ by comparing r(2) with the corresponding value from the same choice function. The correct answer choice will match $r(x)$ for both tested values, indicating it is algebraically equivalent to $p(x)\cdot q(x)$.

We are given

The product $p(x)\cdot q(x)$ means

So our task is to expand and simplify this product of two binomials.

Use the distributive property: multiply each term in the first parentheses by each term in the second parentheses:

Now simplify each of these four products one by one.

Compute each term:

• First term: $\left(\frac{3}{4}x\right)(8x) = \frac{3}{4}\cdot 8\cdot x^2 = 6x^2$.
• Second term: $\left(\frac{3}{4}x\right)(9) = \frac{27}{4}x$.
• Third term: $-\left(\frac{5}{6}\right)(8x) = -\frac{40}{6}x = -\frac{20}{3}x$.
• Fourth term: $-\left(\frac{5}{6}\right)(9) = -\frac{45}{6} = -\frac{15}{2}$.

So after multiplying, we have

Now combine the two $x$ terms, $\frac{27}{4}x$ and $-\frac{20}{3}x$.

Find a common denominator for the fractions $\frac{27}{4}$ and $-\frac{20}{3}$. The least common denominator is $12$:

Add them:

So the combined $x$ term is $\frac{1}{12}x$.

Substitute the combined $x$ term back into the expression:

Compare this with the answer choices; it matches choice A: $6x^2 + \dfrac{1}{12}x - \dfrac{15}{2}$.