To factor the expression \(5x^2 + 12x + 4\) completely, we can use factoring by grouping or the quadratic formula. However, we'll try to factor it directly first.
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Start with the given expression: \[ 5x^2 + 12x + 4 \]
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We look for two numbers that multiply to \(5 \times 4 = 20\) and add up to \(12\). The numbers \(10\) and \(2\) fit because \(10 \times 2 = 20\) and \(10 + 2 = 12\).
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Rewrite the middle term \(12x\) using \(10x\) and \(2x\): \[ 5x^2 + 10x + 2x + 4 \]
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Now, group the terms: \[ (5x^2 + 10x) + (2x + 4) \]
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Factor out the common factors from each group: \[ 5x(x + 2) + 2(x + 2) \]
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Notice that \((x + 2)\) is a common factor: \[ (5x + 2)(x + 2) \]
Thus, the completely factored form of the expression \(5x^2 + 12x + 4\) is: \[ (5x + 2)(x + 2) \]