We need to divide the polynomial \( \frac{x^2 + 4x - 5}{x - 1} \). First, we will factor the numerator \( x^2 + 4x - 5 \).
To factor \( x^2 + 4x - 5 \), we look for two numbers that multiply to \(-5\) (the constant term) and add to \(4\) (the coefficient of the linear term). The numbers \(5\) and \(-1\) work:
\[ x^2 + 4x - 5 = (x + 5)(x - 1). \]
Now we can rewrite the original expression using this factorization:
\[ \frac{x^2 + 4x - 5}{x - 1} = \frac{(x + 5)(x - 1)}{x - 1}. \]
Next, we can cancel the \(x - 1\) terms (as long as \(x \neq 1\) to avoid division by zero):
\[ \frac{(x + 5)(x - 1)}{x - 1} = x + 5 \quad \text{(for } x \neq 1\text{)}. \]
Thus, the final answer is:
\[ \boxed{x + 5}. \]
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