Certainly! Let's solve the expression [tex]\( x^4 + x^2 + 1 \)[/tex] using the hint provided: "Add and subtract [tex]\( x^2 \)[/tex]."
### Step-by-Step Solution
1. Start with the original expression:
[tex]\[
x^4 + x^2 + 1
\][/tex]
2. Add and subtract [tex]\( x^2 \)[/tex]:
[tex]\[
x^4 + x^2 + 1 + x^2 - x^2
\][/tex]
3. Rewrite the expression by grouping terms:
[tex]\[
(x^4 + 2x^2 + 1) - x^2
\][/tex]
4. Observe that [tex]\( x^4 + 2x^2 + 1 \)[/tex] can be written as a perfect square:
[tex]\[
(x^2 + 1)^2
\][/tex]
Therefore, we have:
[tex]\[
(x^2 + 1)^2 - x^2
\][/tex]
5. Final expression:
[tex]\[
x^4 + x^2 + 1 = (x^2 + 1)^2 - x^2
\][/tex]
### Summary
Through the steps above, we have successfully rewritten the original expression by using the given hint. The final form of the expression is:
[tex]\[
x^4 + x^2 + 1 = (x^2 + 1)^2 - x^2
\][/tex]
This form can be particularly useful in various contexts such as integration, differentiation, or solving algebraic equations.
