SAT Question 89fc23af: Equivalent Expressions answer and explanation with Desmos solution

Choice D is correct. The numerator of the given expression can be rewritten in terms of the denominator, $x-3$, as follows: $x^2-2x-5=x^2-3x+x-3-2$, which is equivalent to $x(x-3)+(x-3)-2$. So the given expression is equivalent to $\frac{x(x-3)+(x-3)-2}{x-3}=\frac{x(x-3)}{x-3}+\frac{x-3}{x-3}-\frac{2}{x-3}$. Since the given expression is defined for $x\ne 3$, the expression can be rewritten as $x+1-\frac{2}{x-3}$. Long division can also be used as an alternate approach.