To simplify the expression \(\frac{x^2 + 5x + 6}{x + 2}\), we first factor the numerator.
The expression in the numerator, \(x^2 + 5x + 6\), can be factored as follows:
- We look for two numbers that multiply to \(6\) (the constant term) and add up to \(5\) (the coefficient of the \(x\) term). The numbers \(2\) and \(3\) satisfy these conditions since \(2 \times 3 = 6\) and \(2 + 3 = 5\).
Thus, we can factor the numerator: \[ x^2 + 5x + 6 = (x + 2)(x + 3) \]
Now we can rewrite the original expression: \[ \frac{x^2 + 5x + 6}{x + 2} = \frac{(x + 2)(x + 3)}{x + 2} \]
Since \(x + 2\) is a common factor in the numerator and the denominator, we can cancel it (as long as \(x \neq -2\)): \[ = x + 3 \quad \text{(for } x \neq -2\text{)} \]
Thus, the simplified expression in standard form is: \[ \boxed{x + 3} \]
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