The expression $(x + 1)^6$ can be expanded using the binomial theorem, which is a way to expand powers of binomials.

The binomial expansion for $(a + b)^n$ is given by:

$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

In your case, $a = x$, $b = 1$, and $n = 6$. So, we can apply the binomial expansion to $(x + 1)^6$:

$(x + 1)^6 = \sum_{k=0}^{6} \binom{6}{k} x^{6-k} 1^k$

Now, let's compute the individual terms:

(x + 1)^6 &= \binom{6}{0} x^6 1^0 + \binom{6}{1} x^5 1^1 + \binom{6}{2} x^4 1^2 + \binom{6}{3} x^3 1^3 \\ &\quad + \binom{6}{4} x^2 1^4 + \binom{6}{5} x^1 1^5 + \binom{6}{6} x^0 1^6 \\ &= x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1 \end{aligned}$$ Thus, the expansion of $$(x + 1)^6$$ is: $$x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1$$ Would you like a more detailed explanation or any clarification on the binomial theorem? Here are some related questions for further exploration: 1. What is the binomial theorem and how is it used? 2. How do you expand higher powers like $$(x + 1)^n$$ for large values of $$n$$? 3. What is the general formula for binomial coefficients $$\binom{n}{k}$$? 4. How does the expansion of $$(x - 1)^6$$ differ from $$(x + 1)^6$$? 5. What is the significance of each term in a binomial expansion? **Tip**: When expanding binomials, always pay attention to the powers of $$x$$ and the binomial coefficients, which can be found using Pascal’s triangle or the formula $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$.