Once $x^2 - 16$ is factored, what common denominator can you use for both fractions, and what do you need to multiply the first fraction by to get it?

When you combine the fractions, the expression becomes one big numerator over a common denominator. Make sure you subtract the entire second numerator, not just part of it—distribute the minus sign.

After expanding and distributing, group like terms in the numerator (the $x$ terms and the constant terms). Then compare that simplified numerator to the numerators in the choices.

In Desmos, type the original expression as y = 2/(x-4) - (3x+1)/(x^2-16) so you can see its graph.

On separate lines, enter each choice, for example y = (7-x)/((x-4)(x+4)), y = (x-7)/((x-4)(x+4)), etc. Make sure parentheses match the given denominators exactly.

Look at the graphs: the correct choice will produce a graph that lies exactly on top of the graph of the original expression for all $x$ where the expression is defined. The others will differ in slope or position, especially for positive and negative $x$ values.

You can also pick a few $x$-values (avoiding $x=\pm 4$ where the expression is undefined), use the Desmos table for each function, and see which choice always gives the same $y$-values as the original expression.

Notice that $x^2-16$ is a difference of squares:

So the original expression becomes

To combine the fractions, they need the same denominator. The second fraction already has denominator $(x-4)(x+4)$.

For the first fraction $\dfrac{2}{x-4}$, multiply top and bottom by $(x+4)$:

Now the expression is

Now that the denominators match, subtract the numerators over the common denominator:

First expand $2(x+4)$:

So the numerator becomes

Distribute the minus sign across $(3x+1)$:

Combine like terms in the numerator:

• For the $x$ terms: $2x - 3x = -x$.
• For the constants: $8 - 1 = 7$.

So the numerator simplifies to $-x + 7$, which is the same as $7 - x$.

Therefore the whole expression becomes

which matches answer choice A.