To multiply the binomials \((x-7y)(7x+6y)\) using the FOIL method, we follow these steps:
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First: Multiply the first terms of each binomial. \[ x \cdot 7x = 7x^2 \]
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Outer: Multiply the outer terms of the binomials. \[ x \cdot 6y = 6xy \]
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Inner: Multiply the inner terms of the binomials. \[ -7y \cdot 7x = -49xy \]
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Last: Multiply the last terms of each binomial. \[ -7y \cdot 6y = -42y^2 \]
Now, combine all these results: \[ 7x^2 + 6xy - 49xy - 42y^2 \]
Next, combine the like terms (\(6xy\) and \(-49xy\)): \[ 7x^2 + (6xy - 49xy) - 42y^2 = 7x^2 - 43xy - 42y^2 \]
Thus, the simplified result is: \[ \boxed{7x^2 - 43xy - 42y^2} \]
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