To use the FOIL method on the expression \((x - 1)(x + 3)\), we will multiply the terms in the following manner:
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F (First): Multiply the first terms in each binomial: \[ x \cdot x = x^2 \]
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O (Outside): Multiply the outer terms in the product: \[ x \cdot 3 = 3x \]
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I (Inside): Multiply the inner terms: \[ -1 \cdot x = -x \]
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L (Last): Multiply the last terms in each binomial: \[ -1 \cdot 3 = -3 \]
Now, combine all the results: \[ x^2 + 3x - x - 3 \]
Combine like terms: \[ x^2 + (3x - x) - 3 = x^2 + 2x - 3 \]
Thus, the product is: \[ \boxed{x^2 + 2x - 3} \]