To factor the quadratic expression \(5w^2 + 9w - 2\), we can use the method of factoring by grouping.
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Multiply the leading coefficient and the constant term: \(5 \times (-2) = -10\).
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Find two numbers that multiply to \(-10\) and add to \(9\). The numbers \(10\) and \(-1\) fit these criteria since:
- \(10 \times (-1) = -10\)
- \(10 + (-1) = 9\)
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Rewrite the middle term using these two numbers: \[ 5w^2 + 10w - 1w - 2 \]
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Group the terms: \[ (5w^2 + 10w) + (-1w - 2) \]
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Factor out the common factors from each group:
- From the first group, \(5w\) can be factored out: \[ 5w(w + 2) \]
- From the second group, \(-1\) can be factored out: \[ -1(w + 2) \]
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Combine the factored terms: \[ 5w(w + 2) - 1(w + 2) \] This can be written as: \[ (5w - 1)(w + 2) \]
Therefore, the completely factored form of \(5w^2 + 9w - 2\) is: \[ \boxed{(5w - 1)(w + 2)} \]