Question 105·Medium·Advanced Math

$p(x)\cdot q(x)$ means multiply the two outputs at the same input. Type both function rules in Desmos, then type $p(0)q(0)$. Desmos shows $-2.7$, which equals $-\frac{27}{10}$. At $x=0$, every term containing $x$ vanishes, so the equivalent expression must have constant term $-\frac{27}{10}$.

The $x^2$ term comes from multiplying the $x$ terms in the two given rules. Their coefficients are $\frac{5}{8}$ and $4$, so $\left(\frac{5}{8}x\right)(4x)=\frac{5}{2}x^2$. Both choices still in play have that same coefficient.

The two choices with that constant share the $\frac{5}{2}x^2$ term. Their only remaining difference is the sign of the $x$ term. Type $p(1)q(1)$, then evaluate both remaining expressions at $x=1$ as shown. Desmos gives $-1.925$ for the product, $1.525$ for the positive $x$ term, and $-1.925$ for the negative $x$ term. Because the $x^2$ terms already match, the tests at $0$ and $1$ settle both other coefficients. So $p(x)\cdot q(x)=\frac{5}{2}x^2-\frac{69}{40}x-\frac{27}{10}$. Choice D.

The distributive property says each term in one parentheses multiplies each term in the other. Write the four pairings before combining anything:

Multiply the four pairs, keeping the negative factor attached to both of its products:

Like terms have the same variable part, so the two $x$ terms combine but the $x^2$ term stays separate. Type $\frac{15}{8}-\frac{18}{5}$ in Desmos. It shows $-1.725$; its fraction button shows $-\frac{69}{40}$. Combine the $x$ terms:

Keep the $x^2$ term and the negative constant, then use the combined $x$ term: $p(x)q(x)=\frac{5}{2}x^2-\frac{69}{40}x-\frac{27}{10}$. Choice D.