The problem requires expanding and simplifying the polynomial $p(x) = (x + 2)^4$. Let’s solve it step by step.

Expansion Process:

1. Apply the Binomial Theorem: $(x + 2)^4 = \sum_{k=0}^{4} \binom{4}{k} x^{4-k} \cdot 2^k$ where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.

2. Compute Each Term:

   • For $k = 0$: $\binom{4}{0} x^4 \cdot 2^0 = 1 \cdot x^4 = x^4$
   • For $k = 1$: $\binom{4}{1} x^3 \cdot 2^1 = 4 \cdot x^3 \cdot 2 = 8x^3$
   • For $k = 2$: $\binom{4}{2} x^2 \cdot 2^2 = 6 \cdot x^2 \cdot 4 = 24x^2$
   • For $k = 3$: $\binom{4}{3} x^1 \cdot 2^3 = 4 \cdot x \cdot 8 = 32x$
   • For $k = 4$: $\binom{4}{4} x^0 \cdot 2^4 = 1 \cdot 16 = 16$

3. Combine the Terms: $(x + 2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16$

Final Answer:

$p(x) = x^4 + 8x^3 + 24x^2 + 32x + 16$

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Related Questions:

1. How does the binomial theorem work for higher powers like $(x + y)^n$?
2. What are Pascal’s Triangle and its connection to binomial expansion?
3. Can we apply the same principles to polynomials of three terms, such as $(x + y + z)^4$?
4. How can we use the distributive property instead of the binomial theorem to expand this polynomial?
5. How can this expansion be visualized geometrically?

Tip:

Always double-check binomial coefficients using Pascal's Triangle for small values of $n$, as it’s a quick and reliable tool.