Question 171·Hard·Advanced Math

Choose $x=3$ because $3>2$. Type the given functions into Desmos, then type $q(3)$ to check the denominator and $p(3)/q(3)$ to evaluate the fraction. Desmos shows $q(3)=5$, so division is allowed, and the fraction equals $18$.

Set $x_1=3$, then type each choice with $x_1$ replacing $x$. In the order listed, Desmos shows $30$, $22.8$, $18$, and $3.6$. Only $x^2+5x-6$ matches the original's value. Since every other choice fails at an allowed input, the equivalent expression is $x^2+5x-6$. Choice C.

No Desmos needed. Multiplying by the denominator checks the expression exactly. A quotient is the result of division. The numerator begins $x^4+5x^3$, while the denominator begins $x^2$, so start the quotient with $x^2+5x$: $x^2(x^2+5x)=x^4+5x^3$.

Distribute both terms of $q(x)=x^2-4$ across that partial quotient: $(x^2-4)(x^2+5x)=x^4+5x^3-4x^2-20x$. This gives the first two terms of $p(x)$, but not yet the rest.

Subtract that product from $p(x)$: $\begin{aligned}p(x)-(x^2-4)(x^2+5x)&=(x^4+5x^3-10x^2-20x+24)\\&\quad-(x^4+5x^3-4x^2-20x)\\&=-6x^2+24.\end{aligned}$

Factor $-6$ from the difference: $-6x^2+24=-6(x^2-4)$. That's $-6q(x)$, so $p(x)=q(x)(x^2+5x)-6q(x)$.

Factor out $q(x)$, then cancel it: $\frac{p(x)}{q(x)}=\frac{q(x)(x^2+5x-6)}{q(x)}=x^2+5x-6$. Because $x>2$, $x^2-4=(x-2)(x+2)$ isn't zero, so the cancellation is valid. The equivalent expression is $x^2+5x-6$. Choice C.