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The question asks us to use the discriminant to determine the number and types of solutions for the quadratic equation:

\[
-2x - 2x^2 + 4 = 0
\]

To analyze this, let's rewrite it in the standard form for a quadratic equation:

\[
ax^2 + bx + c = 0
\]

### Step 1: Rewrite in Standard Form
Rearrange the terms:
\[
-2x^2 - 2x + 4 = 0
\]

So here, \(a = -2\), \(b = -2\), and \(c = 4\).

### Step 2: Calculate the Discriminant
The discriminant for a quadratic equation \(ax^2 + bx + c = 0\) is given by:
\[
D = b^2 - 4ac
\]

Substitute \(a = -2\), \(b = -2\), and \(c = 4\):

\[
D = (-2)^2 - 4 \cdot (-2) \cdot 4
\]
\[
D = 4 + 32
\]
\[
D = 36
\]

### Step 3: Determine the Nature of the Solutions
The discriminant value \(D = 36\) is positive. This tells us that:

1. Since \(D > 0\), the equation has **two real and distinct solutions**.

### Answer
The equation has **two real and distinct solutions**.

Would you like further details or have any questions?

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Here are five related questions to expand your understanding:

1. What would change if the discriminant was zero?
2. How do you find the actual solutions of the quadratic equation?
3. What does a negative discriminant indicate about the solutions?
4. How does the discriminant help in graphing a quadratic function?
5. Can you apply the discriminant concept to non-quadratic equations?

**Tip:** Always check that the equation is in standard form \(ax^2 + bx + c = 0\) before finding the discriminant.

Use the discriminant to determine the number and types of solutions of the quadratic equation: -2x - 2x^2 + 4 = 0.

Learn how to use the discriminant to determine the number and types of solutions for the quadratic equation -2x - 2x^2 + 4 = 0. This problem covers essential algebra concepts, including quadratic equations and the discriminant formula, suitable for Grades 9-10.

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