SAT Question 967ef685: Equivalent Expressions answer and explanation with Desmos solution

Choice D is correct. Two fractions can be added together when they have a common denominator. Since $k>0$, multiplying the second term in the given expression by $\frac{k}{k}$ yields $\frac{\left(42ak\right)k}{k}$, which is equivalent to $\frac{42a{k}^2}{k}$. Therefore, the expression $\frac{42a}{k}+42ak$ can be written as $\frac{42a}{k}+\frac{42a{k}^2}{k}$ which is equivalent to $\frac{42a+42a{k}^2}{k}$. Since each term in the numerator of this expression has a factor of $42a$, the expression $\frac{42a+42a{k}^2}{k}$ can be rewritten as $\frac{42a\left(1\right)+42a\left(k^2\right)}{k}$, or $\frac{42a\left(1+k^2\right)}{k}$, which is equivalent to $\frac{42a(k^2+1)}{k}$.