Solve the following inequality. \[ x^{2}+11 x+28<0 \] Select the correct choice below and, if necessary, fill in the answer box. A. The solution set is (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.) B. There is no real solution.

To solve the inequality \( x^{2}+11x+28<0 \), we first need to find the roots of the corresponding equation \( x^{2}+11x+28=0 \). Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \): Here, \( a = 1 \), \( b = 11 \), and \( c = 28 \). Calculating the discriminant: \[ b^2 - 4ac = 11^2 - 4(1)(28) = 121 - 112 = 9. \] Now applying the quadratic formula: \[ x = \frac{-11 \pm \sqrt{9}}{2} = \frac{-11 \pm 3}{2}. \] This gives us the roots: \[ x = \frac{-11 + 3}{2} = -4 \quad \text{and} \quad x = \frac{-11 - 3}{2} = -7. \] Next, we analyze the intervals determined by the roots: \( (-\infty, -7) \), \( (-7, -4) \), and \( (-4, \infty) \). Using test points in each interval: 1. For \( x = -8 \) in \( (-\infty, -7) \): \[ (-8)^2 + 11(-8) + 28 = 64 - 88 + 28 = 4 \quad (\text{not } < 0) \] 2. For \( x = -6 \) in \( (-7, -4) \): \[ (-6)^2 + 11(-6) + 28 = 36 - 66 + 28 = -2 \quad (< 0) \] 3. For \( x = -3 \) in \( (-4, \infty) \): \[ (-3)^2 + 11(-3) + 28 = 9 - 33 + 28 = 4 \quad (\text{not } < 0) \] Thus, the inequality holds in the interval \( (-7, -4) \). The final answer is: A. The solution set is \( (-7, -4) \)