Solve the following inequality: \( \frac{x^{2}-12 x+35}{x^{2}+16 x+64}<0 \) State your answer using interval notation. Use U for union and oo for \( \infty \).

To solve the inequality \( \frac{x^{2}-12 x+35}{x^{2}+16 x+64}<0 \), we first factor the numerator and the denominator. The numerator \( x^{2}-12 x+35 \) can be factored as: \[ (x - 5)(x - 7) \] The denominator \( x^{2}+16 x+64 \) can be written as: \[ (x + 8)(x + 8) = (x + 8)^2 \] So, we rewrite the inequality: \[ \frac{(x - 5)(x - 7)}{(x + 8)^2} < 0 \] Next, we find the critical points by setting the numerator and the denominator equal to zero: - From the numerator: \( x - 5 = 0 \) gives \( x = 5 \) and \( x - 7 = 0 \) gives \( x = 7 \). - From the denominator: \( (x + 8)^2 = 0 \) gives \( x = -8 \) (note that this point will make the expression undefined). Now we will test the sign of the expression in the intervals determined by these critical points: \( (-\infty, -8) \), \( (-8, 5) \), \( (5, 7) \), and \( (7, \infty) \). 1. **Interval \( (-\infty, -8) \)**: Choose \( x = -9 \): \[ \frac{(-9 - 5)(-9 - 7)}{(-9 + 8)^2} = \frac{(-14)(-16)}{1} > 0 \] 2. **Interval \( (-8, 5) \)**: Choose \( x = 0 \): \[ \frac{(0 - 5)(0 - 7)}{(0 + 8)^2} = \frac{(-5)(-7)}{64} > 0 \] 3. **Interval \( (5, 7) \)**: Choose \( x = 6 \): \[ \frac{(6 - 5)(6 - 7)}{(6 + 8)^2} = \frac{(1)(-1)}{196} < 0 \] 4. **Interval \( (7, \infty) \)**: Choose \( x = 8 \): \[ \frac{(8 - 5)(8 - 7)}{(8 + 8)^2} = \frac{(3)(1)}{256} > 0 \] Now we gather the intervals where the inequality is less than zero. This occurs only in \( (5, 7) \). Finally, since \( x = -8 \) is not included (as it makes the denominator zero) and the endpoints \( x = 5 \) and \( x = 7 \) are not included since the inequality is strict (<), we express our solution in interval notation as: \((5, 7)\)