After expanding $(x + y)^3$, try to rearrange the equation so that $x^3 + y^3$ is alone on one side.

Once you have $x^3 + y^3$ written using $x + y$ and $xy$, replace $x + y$ with $r$ and $xy$ with $s$ as given in the problem.

In Desmos, choose simple numbers for $x$ and $y$, such as x = 2 and y = 5. On a new line, enter E = x^3 + y^3 and note the value of E that Desmos shows.

On separate lines, enter r = x + y and s = x * y. Desmos will compute numerical values for $r$ and $s$ using your chosen $x$ and $y$.

On new lines, enter each option as a separate expression, for example A = r^3 - 3*r*s, B = r^3 + 3*r*s, C = (r - s)^3, and D = r^2 - 3*s. Desmos will display a numerical value for each.

Compare the values of A, B, C, and D with the value of E. The option whose value matches E for your chosen $x$ and $y$ is the expression that correctly represents $x^3 + y^3$ in terms of $r$ and $s$.

Start with the binomial expansion of $(x + y)^3$:

Notice that this includes the $x^3$ and $y^3$ terms we want, plus some extra mixed terms.

Group the $x^3$ and $y^3$ terms on one side and the mixed terms together:

Now factor the mixed terms:

So we can rewrite the equation as:

Now solve for $x^3 + y^3$:

We are given that $x + y = r$ and $xy = s$. Substitute these into the expression from the previous step:

Since multiplication is commutative, $3sr = 3rs$, so this becomes

Thus, the expression equal to $x^3 + y^3$ is $r^3 - 3rs$.