To find the product \((x+7y)(7x+6y)\), we will use the distributive property (also known as the FOIL method for binomials).
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Multiply the first terms: \[ x \cdot 7x = 7x^2 \]
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Multiply the outer terms: \[ x \cdot 6y = 6xy \]
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Multiply the inner terms: \[ 7y \cdot 7x = 49xy \]
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Multiply the last terms: \[ 7y \cdot 6y = 42y^2 \]
Now, we will combine all these results: \[ 7x^2 + 6xy + 49xy + 42y^2 \]
Next, we combine like terms (\(6xy\) and \(49xy\)): \[ 7x^2 + (6xy + 49xy) + 42y^2 = 7x^2 + 55xy + 42y^2 \]
Therefore, the simplified product is: \[ \boxed{7x^2 + 55xy + 42y^2} \]
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