To solve the equation:

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

### Step 1: Rearrange the equation

First, move the terms around to make the leading term positive:

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

### Step 2: Substitution

Let \( y = x^2 \), which makes the equation:

\[
y^2 - y - 2 = 0
\]

Now, solve this quadratic equation.

### Step 3: Solve the quadratic equation

The quadratic equation is:

\[
y^2 - y - 2 = 0
\]

Use the quadratic formula:

\[
y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-2)}}{2(1)}
\]

\[
y = \frac{1 \pm \sqrt{1 + 8}}{2}
\]

\[
y = \frac{1 \pm \sqrt{9}}{2}
\]

\[
y = \frac{1 \pm 3}{2}
\]

Thus, the two solutions are:

\[
y = \frac{1 + 3}{2} = 2 \quad \text{and} \quad y = \frac{1 - 3}{2} = -1
\]

### Step 4: Solve for \( x \)

Recall that \( y = x^2 \). Now, solve for \( x \) in both cases:

- For \( y = 2 \):

\[
x^2 = 2 \quad \Rightarrow \quad x = \pm \sqrt{2}
\]

- For \( y = -1 \):

\[
x^2 = -1
\]

There is no real solution for this equation because the square of a real number cannot be negative. Hence, the solutions for \( x^2 = -1 \) are complex:

\[
x = \pm i
\]

### Final Answer:

The solutions are:

\[
x = \pm \sqrt{2} \quad \text{(real solutions)}, \quad x = \pm i \quad \text{(complex solutions)}
\]

Would you like more details or have any questions?

Here are five related questions to explore:
1. How does substitution help simplify higher-degree polynomials?
2. What is the quadratic formula and how is it derived?
3. What are complex numbers, and how are they used in solving equations?
4. How do you identify and interpret real versus complex solutions?
5. What is the importance of the discriminant in a quadratic equation?

**Tip:** The discriminant (\(b^2 - 4ac\)) helps you quickly determine whether a quadratic equation has real or complex solutions.

This detailed solution explains how to solve the polynomial equation -x^4 + x^2 + 2 = 0 using substitution and the quadratic formula. It includes real and complex solutions and explores key algebraic concepts such as polynomials, quadratic equations, and complex numbers. Suitable for students in Grades 9-12.

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