To factor the quadratic expression \( 2g^2 + 9g + 10 \), we can use the method of factoring by grouping after finding two numbers that multiply to \( (2 \times 10 = 20) \) and add up to \( 9 \).
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Identify the product and sum: We need two numbers that multiply to \( 20 \) (the product of the coefficient of \( g^2 \) and the constant term) and add up to \( 9 \) (the coefficient of \( g \)).
The numbers \( 4 \) and \( 5 \) fit this because: \[ 4 \times 5 = 20 \quad \text{and} \quad 4 + 5 = 9. \]
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Rewrite the middle term using these two numbers: \[ 2g^2 + 4g + 5g + 10. \]
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Now group the terms: \[ (2g^2 + 4g) + (5g + 10). \]
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Factor out the greatest common factors from each group: \[ 2g(g + 2) + 5(g + 2). \]
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Now, factor out the common binomial factor \((g + 2)\): \[ (g + 2)(2g + 5). \]
Thus, the completely factored form of \( 2g^2 + 9g + 10 \) is: \[ \boxed{(g + 2)(2g + 5)}. \]