SAT Question 20291f47: Equivalent Expressions answer and explanation with Desmos solution

Choice C is correct. Factoring the denominator in the second term of the given expression gives $\frac{y+12}{x-8}+\frac{y\left(x-8\right)}{xy\left(x-8\right)}$. This expression can be rewritten with common denominators by multiplying the first term by $\frac{xy}{xy}$, giving $\frac{xy\left(y+12\right)}{xy\left(x-8\right)}+\frac{y\left(x-8\right)}{xy\left(x-8\right)}$. Adding these two terms yields $\frac{xy\left(y+12\right)+y\left(x-8\right)}{xy\left(x-8\right)}$. Using the distributive property to rewrite this expression gives $\frac{x{y}^2+12xy+xy-8y}{x^{2}y-8xy}$. Combining the like terms in the numerator of this expression gives $\frac{x{y}^2+13xy-8y}{x^{2}y-8xy}$.