To perform long division with the polynomial \(x^3 - 5x^2 + 7x - 12\) divided by \(x - 4\), follow these steps:
```
x^2 + (-x + 3)
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x - 4 | x^3 - 5x^2 + 7x - 12
- (x^3 - 4x^2)
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-x^2 + 7x
- (-x^2 + 4x)
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3x - 12
- (3x - 12)
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0
```
So, \(x^3 - 5x^2 + 7x - 12\) divided by \(x - 4\) equals \(x^2 - x + 3\).