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Rewrite as f(x)=-x^4 + 6x^3 - 9x^2 (where ^ means to the power of)

= -x^2(x^2 - 6x + 9)

= -x^2 (x -3)(x-3)

So the zeroes are 0, 0, 3, 3.

I am not sure what you mean by multiplicities.

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