To solve the inequality \( x(x + 9) + 24 > 17 \), follow these steps:

1. **Simplify the inequality**:
   \[
   x(x + 9) + 24 > 17
   \]
   First, expand the expression:
   \[
   x^2 + 9x + 24 > 17
   \]

2. **Move all terms to one side**:
   Subtract 17 from both sides:
   \[
   x^2 + 9x + 24 - 17 > 0
   \]
   Simplify:
   \[
   x^2 + 9x + 7 > 0
   \]

3. **Solve the corresponding quadratic equation**:
   Solve \( x^2 + 9x + 7 = 0 \) using the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   Here, \( a = 1 \), \( b = 9 \), and \( c = 7 \).

   Calculate the discriminant:
   \[
   \text{Discriminant} = b^2 - 4ac = 9^2 - 4(1)(7) = 81 - 28 = 53
   \]
   Since the discriminant is positive, there are two real roots:
   \[
   x = \frac{-9 \pm \sqrt{53}}{2}
   \]

4. **Find the roots**:
   \[
   x_1 = \frac{-9 + \sqrt{53}}{2}, \quad x_2 = \frac{-9 - \sqrt{53}}{2}
   \]
   These are the roots of the quadratic equation.

5. **Determine the intervals where the inequality holds**:
   The quadratic \( x^2 + 9x + 7 \) opens upwards (since the coefficient of \( x^2 \) is positive), so the inequality \( x^2 + 9x + 7 > 0 \) holds for \( x \) values outside the interval defined by the roots. Therefore, the solution to the inequality is:
   \[
   x < \frac{-9 - \sqrt{53}}{2} \quad \text{or} \quad x > \frac{-9 + \sqrt{53}}{2}
   \]

### Final Answer:
\[
x < \frac{-9 - \sqrt{53}}{2} \quad \text{or} \quad x > \frac{-9 + \sqrt{53}}{2}
\]

Would you like more details on any of these steps, or do you have any questions?

### Related Questions:
1. How do you solve a quadratic equation by factoring?
2. What are the properties of the discriminant in the quadratic formula?
3. Can a quadratic inequality ever have no real solutions?
4. How does the sign of the leading coefficient in a quadratic equation affect the graph?
5. What are the methods to solve a quadratic inequality?

### Tip:
When solving inequalities, always remember to consider the sign of the quadratic expression and the intervals between the roots.

Learn how to solve the quadratic inequality x(x + 9) + 24 > 17 with a detailed step-by-step solution. This problem covers key algebraic concepts, including solving quadratic inequalities and using the quadratic formula. Suitable for students in Grades 9-11.

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