The equation given is:

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

Let's solve this quadratic equation step by step.

### Step 1: Move all terms to one side of the equation
Add \( 4 \) to both sides to set the equation to zero:

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

### Step 2: Apply the quadratic formula
The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \( 2x^2 - x + 4 = 0 \), the coefficients are:
- \( a = 2 \)
- \( b = -1 \)
- \( c = 4 \)

### Step 3: Calculate the discriminant
The discriminant \( \Delta \) is:

\[
\Delta = b^2 - 4ac = (-1)^2 - 4(2)(4) = 1 - 32 = -31
\]

Since the discriminant is negative (\( \Delta = -31 \)), the equation has no real solutions, but it has two complex solutions.

### Step 4: Find the complex solutions
Substitute the values into the quadratic formula:

\[
x = \frac{-(-1) \pm \sqrt{-31}}{2(2)} = \frac{1 \pm \sqrt{31}i}{4}
\]

So the solutions are:

\[
x = \frac{1 + \sqrt{31}i}{4} \quad \text{and} \quad x = \frac{1 - \sqrt{31}i}{4}
\]

These are the two complex solutions to the equation.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How do you determine the nature of the roots using the discriminant?
2. What are the implications of having a negative discriminant in a quadratic equation?
3. How can you graph a quadratic function with complex roots?
4. What is the geometric interpretation of complex solutions?
5. How do you convert a quadratic equation to its vertex form?

**Tip:** When dealing with complex solutions, remember that they always come in conjugate pairs for real-coefficient polynomials.

Explore how to solve the quadratic equation 2x^2 - x = -4 step-by-step, including the application of the quadratic formula and understanding complex solutions. This problem deals with quadratic equations, complex numbers, and the discriminant. Suitable for high school students.

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