To divide the polynomials \(\frac{x^2 + 7x + 10}{x + 5}\), we will first factor the numerator.
- The polynomial in the numerator, \(x^2 + 7x + 10\), can be factored. We are looking for two numbers that multiply to \(10\) (the constant term) and add to \(7\) (the coefficient of the \(x\) term). The numbers \(5\) and \(2\) satisfy these conditions since:
- \(5 \times 2 = 10\)
- \(5 + 2 = 7\)
Thus, we can factor the numerator: \[ x^2 + 7x + 10 = (x + 5)(x + 2) \]
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Now we substitute the factored form back into the division: \[ \frac{x^2 + 7x + 10}{x + 5} = \frac{(x + 5)(x + 2)}{x + 5} \]
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We can now cancel the common factor \((x + 5)\) from the numerator and the denominator (assuming \(x \neq -5\) to avoid division by zero): \[ = x + 2 \]
Therefore, the result of the division is: \[ \boxed{x + 2} \]