Try to write $x^3 - 1$ as a product involving $(x - 1)$ and $x^2 - 1$ as a product involving $(x + 1)$. Then see what cancels with the denominators.

After canceling common factors with the denominators, you should be left with a simple expression minus another simple expression. Carefully distribute the subtraction and combine like terms.

Once you have a simplified expression, plug in an easy value like $x = 2$ into both your expression and the original one to confirm they match.

In Desmos, type the original expression as a function, for example: f(x) = (x^3 - 1)/(x - 1) - (x^2 - 1)/(x + 1) (making sure not to use $x = 1$ or $x = -1$ when evaluating).

On new lines, enter g1(x) = x^2 - 2, g2(x) = x + 2, g3(x) = x^2 + x - 1, and g4(x) = x^2 + 2.

Look at the graphs of $f(x)$ and each $g_i(x)$. The correct choice will have a graph that lies exactly on top of the graph of $f(x)$ for all $x$ values where $f(x)$ is defined (excluding $x = 1$ and $x = -1$).

For extra confirmation, use Desmos’s table feature for $f(x)$ and each $g_i(x)$, and compare their values at several $x$ values such as $x = 0, 2, 3$. The function whose values always match those of $f(x)$ is the equivalent expression.

Notice that the expression is

with numerators $x^3 - 1$ and $x^2 - 1$.

These match special factoring patterns:

• $x^3 - 1$ is a difference of cubes.
• $x^2 - 1$ is a difference of squares.

Factor each one:

Now substitute the factored forms into each fraction:

Since $x \neq 1$, you can cancel the common factor $(x - 1)$:

Similarly,

Since $x \neq -1$, cancel $(x + 1)$:

So the entire expression becomes a difference of two polynomials:

Start with

Distribute the minus sign over the parentheses:

Now combine like terms:

• The $x$ and $-x$ cancel.
• The constants $1 + 1$ add to $2$.

So the expression simplifies to

Among the choices, this is answer choice D, $x^{2} + 2$.