Use the Binomial Theorem to expand and then simplify the result: (x² +x+ 1)³.

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Recognize that the Binomial Theorem applies to expressions of the form $(a + b)^n$. Since the given expression is $(x^2 + x + 1)^3$, which has three terms, we cannot directly apply the Binomial Theorem. Instead, we will expand it by multiplying the expression by itself three times: $(x^2 + x + 1)(x^2 + x + 1)(x^2 + x + 1)$.

First, multiply the first two polynomials: $(x^2 + x + 1)(x^2 + x + 1)$. Use the distributive property (FOIL method) to multiply each term in the first polynomial by each term in the second polynomial.

After multiplying, combine like terms to simplify the result of the first multiplication. This will give you a quadratic expression in terms of $x$.

Next, multiply the simplified result from step 3 by the third polynomial $(x^2 + x + 1)$, again using the distributive property to multiply each term.

Finally, combine like terms from the multiplication in step 4 to get the fully expanded and simplified form of $(x^2 + x + 1)^3$.

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