To simplify the expression \((14x + 6y) + (5x + 12y)(x + y)\), we first need to perform the multiplication and then add the like terms.
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Multiply \((5x + 12y)\) and \((x + y)\): \[ (5x + 12y)(x + y) = 5x \cdot x + 5x \cdot y + 12y \cdot x + 12y \cdot y \] This results in: \[ 5x^2 + 5xy + 12xy + 12y^2 \] Combining the like terms \(5xy + 12xy\): \[ 5x^2 + 17xy + 12y^2 \]
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Now substitute this back into the original expression: \[ (14x + 6y) + (5x^2 + 17xy + 12y^2) \]
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Combine all terms: \[ 5x^2 + 14x + 17xy + 6y + 12y^2 \]
Putting it all together, the expression simplifies to: \[ 5x^2 + 14x + 17xy + 12y^2 + 6y \]
In this expression:
- The coefficient of \(x^2\) is 5,
- The coefficient of \(x\) is 14,
- The coefficient of \(xy\) is 17,
- The coefficient of \(y\) is 6,
- The coefficient of \(y^2\) is 12.
Thus, the correct coefficients are:
- \(5\) for \(x^2\)
- \(14\) for \(x\)
- \(17\) for \(xy\)
- \(6\) for \(y\)
- \(12\) for \(y^2\)
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