To simplify the expression \(\frac{x^2 + 12x + 35}{3x + 15}\), we will first factor both the numerator and the denominator.
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Factor the numerator \(x^2 + 12x + 35\): We look for two numbers that multiply to \(35\) (the constant term) and add to \(12\) (the coefficient of \(x\)). The numbers \(5\) and \(7\) work because: \[ 5 \times 7 = 35 \quad \text{and} \quad 5 + 7 = 12. \] Thus, we can factor the numerator as: \[ x^2 + 12x + 35 = (x + 5)(x + 7). \]
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Factor the denominator \(3x + 15\): We can factor out a \(3\): \[ 3x + 15 = 3(x + 5). \]
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Rewrite the original expression using the factored forms: \[ \frac{x^2 + 12x + 35}{3x + 15} = \frac{(x + 5)(x + 7)}{3(x + 5)}. \]
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Cancel the common factor \((x + 5)\), assuming \(x \neq -5\): \[ = \frac{x + 7}{3}. \]
Thus, the simplified form of your original expression is: \[ \frac{x + 7}{3}. \]
Final Answer: \(\frac{x + 7}{3}\)
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