To simplify the equation \(- (2x + 6)(3x + 4) = 5x^2 - 7x\), we need to first expand the left side and then combine like terms.
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Expand the left side: \[
- (2x + 6)(3x + 4) \] First, calculate \((2x + 6)(3x + 4)\) using the distributive property (also known as the FOIL method for binomials): \[ 2x \cdot 3x + 2x \cdot 4 + 6 \cdot 3x + 6 \cdot 4 \] This simplifies to: \[ 6x^2 + 8x + 18x + 24 = 6x^2 + 26x + 24 \]
Now, applying the negative sign: \[
- (6x^2 + 26x + 24) = -6x^2 - 26x - 24 \]
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Now we have the equation: \[ -6x^2 - 26x - 24 = 5x^2 - 7x \]
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To bring all terms to one side, add \(6x^2 + 26x + 24\) to both sides: \[ 0 = 5x^2 - 7x + 6x^2 + 26x + 24 \]
Combining like terms: \[ 0 = (5x^2 + 6x^2) + (-7x + 26x) + 24 \] \[ 0 = 11x^2 + 19x + 24 \]
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Rearrange it: \[ 11x^2 + 19x + 24 = 0 \]
The simplified expression for the left side gives us a quadratic equation: \[ 11x^2 + 19x + 24 = 0 \]
To summarize, the final result is: \[ 11x^2 + 19x + 24 = 0 \]
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