Once you factor the numerator and denominator of the first fraction, check whether a common factor appears in both so you can cancel it before doing any subtraction.

After simplifying the first fraction, notice what the denominators of the two fractions are. How can you combine them into a single fraction so you only subtract the numerators?

When you subtract the second fraction, make sure the minus sign is distributed across both terms in the numerator $2x+3$, not just the first term.

In Desmos, type the original expression as a function, for example:

y1 = (3x^2 - 2x - 1)/(x^2 - 1) - (2x + 3)/(x + 1)

This will graph the given expression.

On new lines, enter each choice as its own function:

• y2 = (x - 2)/(x + 1)
• y3 = (x - 2)/(x - 1)
• y4 = (x + 2)/(x + 1)
• y5 = (x - 2)/(x^2 - 1)

You will see four additional graphs.

Zoom out or adjust the viewing window so you can clearly see all graphs. The correct choice will have a graph that lies exactly on top of the graph of y1 for all $x$ where both are defined (there may be holes at $x=\pm1$). You can also click the gear icon for each function and add a table to compare $y$-values of y1 and each candidate at several $x$-values; the equivalent expression will match y1's values every time.

First, factor the denominator $x^2-1$ as a difference of squares:

Now factor the numerator $3x^2-2x-1$. You want two numbers that multiply to $3\cdot(-1)=-3$ and add to $-2$. Those numbers are $1$ and $-3$.

Rewrite and factor by grouping:

So the first fraction becomes

In

the factor $(x-1)$ appears in both the numerator and the denominator, so for $x\neq 1$ you can cancel it:

Now rewrite the entire expression using this simplification:

Both fractions now have the same denominator $x+1$, so you can combine them into a single fraction by subtracting the numerators:

Be sure to keep the parentheses so the minus sign applies to the whole second numerator.

Simplify the numerator:

So the entire expression simplifies to

which matches answer choice A.