Expanding and Simplifying (x+1)^4 + (x^2 + x + 1)^2
This article will explore the expansion and simplification of the algebraic expression (x+1)^4 + (x^2 + x + 1)^2.
Expanding the Expressions
We can start by expanding each term individually.

(x+1)^4: This can be expanded using the binomial theorem or by repeated multiplication:
 Binomial Theorem: (x+1)^4 = 1x^4 + 4x^31 + 6x^21^2 + 4x1^3 + 11^4 = x^4 + 4x^3 + 6x^2 + 4x + 1
 Repeated Multiplication: (x+1)^4 = (x+1)(x+1)(x+1)(x+1) = (x^2 + 2x + 1)(x^2 + 2x + 1) = x^4 + 4x^3 + 6x^2 + 4x + 1

(x^2 + x + 1)^2: This can be expanded by squaring the binomial:
 (x^2 + x + 1)^2 = (x^2 + x + 1)(x^2 + x + 1) = x^4 + 2x^3 + 3x^2 + 2x + 1
Combining the Expanded Terms
Now that we have expanded both terms, we can combine them:
(x+1)^4 + (x^2 + x + 1)^2 = (x^4 + 4x^3 + 6x^2 + 4x + 1) + (x^4 + 2x^3 + 3x^2 + 2x + 1)
Simplifying the Expression
Finally, we can simplify the expression by combining like terms:
(x^4 + 4x^3 + 6x^2 + 4x + 1) + (x^4 + 2x^3 + 3x^2 + 2x + 1) = 2x^4 + 6x^3 + 9x^2 + 6x + 2
Conclusion
Therefore, the simplified form of (x+1)^4 + (x^2 + x + 1)^2 is 2x^4 + 6x^3 + 9x^2 + 6x + 2.