Factor the expression completely.
2, minus, x, to the power 4 , minus, x, squared
2−x4−x2

Let's factor the expression \( 2 - x^4 - x^2 \) step by step.

### Step 1: Rewrite the expression
Rearrange the terms for clarity:
\[
2 - x^2 - x^4 = 2 - x^2(1 + x^2)
\]

### Step 2: Let substitution \( y = x^2 \)
Let’s substitute \( y = x^2 \) to simplify the expression:
\[
2 - y - y^2
\]
Now we have a quadratic expression in terms of \( y \):
\[
- y^2 - y + 2
\]

### Step 3: Factor the quadratic
We factor the quadratic:
\[
- (y^2 + y - 2)
\]
Now factor \( y^2 + y - 2 \):
\[
y^2 + y - 2 = (y - 1)(y + 2)
\]
So, the expression becomes:
\[
- (y - 1)(y + 2)
\]

### Step 4: Substitute back \( y = x^2 \)
Substitute \( y = x^2 \) back into the factored form:
\[
- (x^2 - 1)(x^2 + 2)
\]

### Step 5: Factor \( x^2 - 1 \)
Notice that \( x^2 - 1 \) is a difference of squares, so we can factor it further:
\[
x^2 - 1 = (x - 1)(x + 1)
\]
Thus, the fully factored expression is:
\[
- (x - 1)(x + 1)(x^2 + 2)
\]

### Final Answer:
\[
- (x - 1)(x + 1)(x^2 + 2)
\]

Do you need further details on any step, or do you have any questions?

Here are 5 related questions to extend your understanding:
1. How do you factor the sum of cubes?
2. How do you solve quadratic equations by factoring?
3. What is the difference between a perfect square trinomial and a difference of squares?
4. How do you simplify polynomials before factoring?
5. What are complex roots in polynomials?

**Tip:** Always check if you can factor expressions further by using formulas like the difference of squares or sum/product of cubes.

Factor the expression completely: 2 - x^4 - x^2

Learn how to completely factor the expression 2 - x^4 - x^2 using algebraic techniques, including factoring by substitution and the difference of squares. This problem covers advanced algebra concepts suitable for students in Grades 9-11.

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