First question:
I urge you to perform the division using the synthetic division method:
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-4 / 1 3 -6 -6 8
-4 4 8 -8
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1 -1 -2 2 0
Note that there is no remainder. When this is the case, the divisor (here, that's -4) is a root of the given polynomial, and the value of that polynomial, g(-4), is 0.
If the remainder were not 0, then the remainder represents the value of the polynomial for that particular divisor. For example, if x = -3, the remainder is -28. We'd write that as g(-3) = -28.
But here, g(-4) = 0.