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The given cubic equation is:

\[
-4x^3 - 7x^2 + 18x + 9 = 0
\]

and it is factored as:

\[
(x + 3)(-4x^2 + 5x + 3) = 0
\]

To find the discriminant and number of real solutions for this equation, follow these steps:

1. **First Solution (from the linear factor)**:
   \[
   x + 3 = 0 \quad \Rightarrow \quad x = -3
   \]

2. **Second and Third Solutions (from the quadratic factor)**:
   The quadratic part is:
   \[
   -4x^2 + 5x + 3 = 0
   \]
   The discriminant (\(\Delta\)) for a quadratic equation \(ax^2 + bx + c = 0\) is given by:
   \[
   \Delta = b^2 - 4ac
   \]
   For this quadratic, \(a = -4\), \(b = 5\), and \(c = 3\). Substituting these values:
   \[
   \Delta = 5^2 - 4(-4)(3) = 25 + 48 = 73
   \]

Since the discriminant is positive (\(\Delta = 73\)), this means the quadratic equation has two distinct real roots.

### Number of real solutions:
- The linear factor gives 1 real solution (\(x = -3\)).
- The quadratic factor gives 2 real solutions since the discriminant is positive.

Thus, the total number of real solutions is **3**.

### Final answers:
- **Discriminant** = 73
- **Number of real solutions** = 3

Would you like further details or clarifications?

Here are 5 related questions to further your understanding:
1. What is the relationship between the discriminant and the nature of the roots?
2. How do we factorize a cubic equation like this one?
3. Can you explain how the quadratic formula works for complex roots?
4. Why does the discriminant being positive indicate real roots?
5. What is the difference between solving linear and quadratic factors?

**Tip**: Always check the signs of coefficients carefully when using the quadratic formula, as small mistakes in sign can lead to incorrect results.

Given that -4x^3 - 7x^2 + 18x + 9 can be factorized into (x + 3)(-4x^2 + 5x + 3), find the discriminant and determine the number of real solutions for -4x^3 - 7x^2 + 18x + 9 = 0.

This detailed guide shows how to solve the cubic equation -4x^3 - 7x^2 + 18x + 9 using factorization. Learn how to find the discriminant and determine the number of real solutions step by step. Suitable for students in Grades 9-12.

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