To factor the quadratic expression \(2x^2 + 11x + 9\), we can use the method of factoring by grouping or look for two numbers that multiply to \(2 \cdot 9 = 18\) and add to \(11\).
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Identify the product and the sum:
- Product: \(2 \cdot 9 = 18\)
- Sum: \(11\)
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The two numbers that multiply to \(18\) and add up to \(11\) are \(2\) and \(9\).
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Rewrite the middle term (11x) using these two numbers: \[ 2x^2 + 2x + 9x + 9 \]
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Now, group the terms: \[ (2x^2 + 2x) + (9x + 9) \]
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Factor out the common factors in each group: \[ 2x(x + 1) + 9(x + 1) \]
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Notice that \((x + 1)\) is a common factor: \[ (2x + 9)(x + 1) \]
So, the fully factored form of \(2x^2 + 11x + 9\) is: \[ (2x + 9)(x + 1) \]