To divide -3x^3 - 4x^2 + 4x + 3 by x - 3, we can use polynomial long division.
-3x^2 - 13x - 35
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x - 3 | -3x^3 - 4x^2 + 4x + 3
-(-3x^3 + 9x^2)
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-13x^2 + 4x
-(-13x^2 + 39x)
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-35x + 3
-(-35x + 105)
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-102
Therefore, the answer is -3x^2 - 13x - 35, with a remainder of -102.