To factor the expression \(2x^2 + 7x - 30\), we can use the method of factoring by grouping.
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Multiply the coefficient of \(x^2\) (which is 2) by the constant term (which is -30): \[ 2 \times -30 = -60 \]
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Next, we need to find two numbers that multiply to -60 and add to the coefficient of \(x\) (which is 7). The numbers are 12 and -5, since: \[ 12 \times -5 = -60 \quad \text{and} \quad 12 + (-5) = 7 \]
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Rewrite the middle term (7x) using these two numbers: \[ 2x^2 + 12x - 5x - 30 \]
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Group the terms: \[ (2x^2 + 12x) + (-5x - 30) \]
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Factor out the common factors in each group: \[ 2x(x + 6) - 5(x + 6) \]
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Now, factor out the common binomial factor \((x + 6)\): \[ (2x - 5)(x + 6) \]
Thus, the factored form of \(2x^2 + 7x - 30\) is: \[ (2x - 5)(x + 6) \]
The correct response is: \[ \text{left parenthesis 2 x minus 5 right parenthesis left parenthesis x plus 6 right parenthesis} \]
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