To address the given problem, let's carefully analyze and manipulate the expression [tex]\( x^4 + x^2 + 1 \)[/tex]. We are given a hint to add and subtract [tex]\( x^2 \)[/tex] within the expression. Here’s a detailed step-by-step solution:
1. Write down the original expression:
[tex]\[
x^4 + x^2 + 1
\][/tex]
2. Add and subtract [tex]\( x^2 \)[/tex] within the original expression:
[tex]\[
x^4 + x^2 + 1 - x^2 + x^2
\][/tex]
3. Observe that this manipulation does not change the value of the expression. Here, we essentially introduced [tex]\( -x^2 \)[/tex] and [tex]\( +x^2 \)[/tex] which cancel each other out, yielding the same expression:
[tex]\[
x^4 + x^2 + 1 - x^2 + x^2 = x^4 + x^2 + 1
\][/tex]
Therefore, after rewriting the expression by following the hint and adding/subtracting [tex]\( x^2 \)[/tex], the expression simplifies back to its original form.
Thus, the final step shows that the given expression [tex]\( x^4 + x^2 + 1 \)[/tex] rewritten with the addition and subtraction of [tex]\( x^2 \)[/tex] remains:
[tex]\[
x^4 + x^2 + 1 = x^4 + x^2 + 1 - x^2 + x^2
\][/tex]
This reaffirms that the expression [tex]\( x^4 + x^2 + 1 \)[/tex] can successfully be rewritten as [tex]\( x^4 + x^2 + 1 - x^2 + x^2 \)[/tex] without altering its value.
