Question 90·Hard·Advanced Math

Set the denominator to zero to find the excluded input: $x-2=0\quad\Longrightarrow\quad x=2$. The fraction is undefined there. One disagreement at an allowed input disproves equivalence.

Choose $x_1=1$: it's allowed, and $x_1-2=-1$. Type $x_1=1$, then type the original fraction with $x_1$ in place of $x$. Desmos prints $8$. Using $x_1$ gives Desmos a number to evaluate rather than an $x$ to graph.

Keep those lines and type each choice with $x_1$ in place of $x$. Desmos prints $-2$, $-6$, $10$, and $8$, in choice order. The first three disagree with the original. A match at one input alone isn't proof, but after those three are ruled out, the remaining equivalent expression is $2x^2-x+2-\frac{5}{x-2}$, for $x\ne2$. Choice D.

The quotient is the polynomial part after division. To check the surviving form, multiply $2x^2-x+2$ by the given denominator:

Combine terms with the same power of $x$:

The given numerator ends in $-9$, but this product ends in $-4$. Subtract $5$ to get the given numerator: $2x^3-5x^2+4x-4-5=2x^3-5x^2+4x-9$. So the remainder, the part left after division, is $-5$, not $5$.

For $x\ne2$, split the numerator into two fractions with the same denominator:

Cancel the whole factor $x-2$ in the first fraction, but keep $x=2$ excluded: $\frac{(x-2)(2x^2-x+2)}{x-2}-\frac{5}{x-2}=2x^2-x+2-\frac{5}{x-2}$. This matches the original at every allowed input. Choice D.