If m does not equal 2 or -3, what is the value of m if...

Another strategy that will work here is to pick a number for m, and work out what the expression equals for that value of m. So for example, you could plug in m = 1. You'd then find, after some annoying calculation, that the expression equals -1/2.

Then when you plug m=1 into each answer choice, the right answer will also need to be equal to -1/2. If you do that, only answer A gives you -1/2, so it must be correct. One potential danger with this approach is that two answers, by pure coincidence, might both give you -1/2 for your chosen value of m. Then you need to pick another value of m and redo all the work to choose between those two answers.

I would personally find all that number-plugging a lot slower than doing the algebra, but for some test takers it will be faster; it really depends how comfortable you are with the algebraic manipulation the question requires. It's at least worth knowing as a fallback strategy if you don't see any algebraic simplification quickly. To use that strategy well, you need to be able to zero in quickly on a number that will be convenient to calculate with. Often 0 and 1 are good choices, when they're permitted (in the question above, even though the question fails to mention the fact, m cannot be 0, because 'm' appears in a denominator). But in some cases, 0 and 1 are bad choices. So in this question:

\(\dfrac{(m - 72)^2 + (m - 72)}{(m - 71)} =\)

\(\begin{align}\\
&\text{A) } m^2 - 72^2 \\\\
&\text{B) } m^2 - 144m - 72 \\\\
&\text{C) } m - 71 \\\\
&\text{D) } m - 72 \\\\
&\text{E) } m - 73 \\
\end{align}\)

you'd be making your life more difficult than necessary by plugging in m=1, because then you'd get things like (-71)^2 in the numerator. It would be better to plug in something like m=73, so that the expressions (m-72) and (m-71) in the question turn out to be equal to easy numbers. Notice though that some answer choices (B especially) are going to be a bit irritating to plug numbers into no matter what number you choose.

I'd still use algebra in the question above - if you notice you can factor out (m-72) from both of the terms in the numerator you get the answer almost instantly:

\(\begin{align}\\
\frac{(m - 72)^2 + (m - 72)}{(m - 71)} &= \frac{(m-72)[ (m - 72) + 1]}{m-71} \\\\
&= \frac{(m-72)(m-71)}{m - 71} \\\\
&= m - 72\\
\end{align}\)