To factor the quadratic expression \(2g^2 + 7g + 5\), we can use the method of factoring by grouping.
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Multiply the leading coefficient (2) by the constant term (5): \[ 2 \times 5 = 10 \]
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Find two numbers that multiply to 10 (the product) and add to 7 (the middle coefficient). The numbers 5 and 2 fit this requirement: \[ 5 \times 2 = 10 \quad \text{and} \quad 5 + 2 = 7 \]
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Rewrite the middle term (7g) using these two numbers: \[ 2g^2 + 5g + 2g + 5 \]
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Group the terms: \[ (2g^2 + 5g) + (2g + 5) \]
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Factor out the common factors in each group: \[ g(2g + 5) + 1(2g + 5) \]
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Now, factor out the common binomial factor \((2g + 5)\): \[ (2g + 5)(g + 1) \]
Thus, the completely factored form of \(2g^2 + 7g + 5\) is: \[ \boxed{(2g + 5)(g + 1)} \]