So straight up, I first got this by just plugging in all the numbers, but that takes WAY too long.

The easier way to do this is to break down the problem and remember that ***you do not have to simplify this beast all the way.***

So here's the broken down problem, simplifying the given problem just a touch:

(-1)^(-4n) * 3^(-4n) > 3^(6-n)

Ignore the first term for a second. Given that 3^(-4n) & 3^(6-n) both have the same base, you can deduce that as long as -4n > 6-n, the inequality holds.

Now, look at the first term: (-1)^(-4n). If this term yields a negative value for some value "n", then the inequality won't hold. So, it has to be positive. Remember that for (-1)^[integer], if the exponent is an odd integer, then the value will be negative and the inequality won't hold. If that exponent is an even integer, then the value will be positive, and the inequality WILL hold.

So, for this inequality to hold, two things have to be true: 1) the first term's exponent has to be an even positive integer and 2) the second term's exponent has to be greater than the exponent on the other side of the equation.

That translates to:
1)
-4n --> n must be a value such that this exponent is an even positive integer
2)
-4n > 6-n
-3n > 6
n < -2 [remember, you have to flip the inequality sign around when you multiply or divide by a negative number in an inequality!]

So, -5 and -3, answers A and B, are the ones that qualify here. And, when you multiply them by -4, you end up with an even, positive integer for both.

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