Look at the factored denominators. What single product of factors could serve as a common denominator for both fractions?

After you rewrite both fractions with the common denominator, combine their numerators into one expression, then see if you can factor that new numerator.

Once you have a single simplified fraction, compare both its numerator and denominator (in factored form) to each answer option to find an exact match.

In Desmos, type the original expression as a function, for example f(x) = (3x^2 - 12)/(x^2 - 5x - 6) - (2x - 4)/(x + 2). Be careful with parentheses so each numerator and denominator is grouped correctly.

Type four more functions, one for each option, such as g(x) = (x-2)(x^2+10x-24)/((x-6)(x+1)(x+2)), h(x) = (x+2)(x^2+22x+24)/((x-6)(x+1)(x+2)), etc. Make sure the factored forms are entered exactly as written in the choices.

Use the table feature (click the gear next to each function and select 'Table') to see values of $f(x)$ and each choice at several $x$-values (avoid $x = 6, -1, -2$, which make denominators zero). The correct choice will have the same $y$-values as $f(x)$ for all tested $x$.

Look at the graphs of $f(x)$ and each answer choice. The correct expression will produce a graph that lies exactly on top of the graph of $f(x)$ everywhere it is defined, including having the same vertical asymptotes and holes.

Start with the given expression:

Factor each part:

• $3x^2 - 12 = 3(x^2 - 4) = 3(x-2)(x+2)$
• $x^2 - 5x - 6 = (x-6)(x+1)$ (since $-6 \cdot 1 = -6$ and $-6 + 1 = -5$)
• $2x - 4 = 2(x-2)$

So the expression becomes

The denominators are $(x-6)(x+1)$ and $(x+2)$, so a common denominator is

Rewrite each fraction with this denominator:

• First fraction (multiply top and bottom by $(x+2)$):

• Second fraction (multiply top and bottom by $(x-6)(x+1)$):

Now the whole expression is

Since the denominators are the same, subtract the numerators:

Factor out the common $(x-2)$ from the numerator:

Now simplify inside the brackets.

• First part:

• Second part:

Subtract:

Substitute the simplified result back into the factored form:

This matches answer choice D, so the equivalent expression is