Simplify the expression completely if possible. (x^2+7x+12)/(x^2)-9

To simplify the expression \(\frac{x^2 + 7x + 12}{x^2 - 9}\), we need to factor both the numerator and the denominator.<br /><br />First, let's factor the numerator \(x^2 + 7x + 12\):<br />We look for two numbers that multiply to 12 and add up to 7. These numbers are 3 and 4.<br />So, we can factor the numerator as:<br />\[ x^2 + 7x + 12 = (x + 3)(x + 4) \]<br /><br />Next, let's factor the denominator \(x^2 - 9\):<br />This is a difference of squares, which can be factored as:<br />\[ x^2 - 9 = (x + 3)(x - 3) \]<br /><br />Now, we can rewrite the original expression using these factored forms:<br />\[ \frac{x^2 + 7x + 12}{x^2 - 9} = \frac{(x + 3)(x + 4)}{(x + 3)(x - 3)} \]<br /><br />We see that \((x + 3)\) is a common factor in both the numerator and the denominator. We can cancel this common factor:<br />\[ \frac{(x + 3)(x + 4)}{(x + 3)(x - 3)} = \frac{x + 4}{x - 3} \]<br /><br />Thus, the simplified form of the expression is:<br />\[ \boxed{\frac{x + 4}{x - 3}} \]<br /><br />Note: The simplification is valid as long as \(x \neq -3\) and \(x \neq 3\) because these values would make the original denominator zero, which is undefined.