To expand $(x + h)^4$, you can use the binomial theorem. The binomial theorem states that:

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

In this case, $a = x$, $b = h$, and $n = 4$. So,

$(x + h)^4 = \sum_{k=0}^{4} \binom{4}{k} x^{4-k} h^k$

Let's expand this step by step:

$(x + h)^4 = \binom{4}{0} x^4 h^0 + \binom{4}{1} x^3 h^1 + \binom{4}{2} x^2 h^2 + \binom{4}{3} x^1 h^3 + \binom{4}{4} x^0 h^4$

Now calculate each binomial coefficient:

$\binom{4}{0} = 1, \quad \binom{4}{1} = 4, \quad \binom{4}{2} = 6, \quad \binom{4}{3} = 4, \quad \binom{4}{4} = 1$

Substituting these into the expansion:

$(x + h)^4 = 1 \cdot x^4 + 4 \cdot x^3 h + 6 \cdot x^2 h^2 + 4 \cdot x h^3 + 1 \cdot h^4$

Simplifying, we get:

$(x + h)^4 = x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4$

Final Answer:

$(x + h)^4 = x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4$

Would you like to explore this expansion further, or do you have any other questions?

Related Questions:

1. How does the expansion change for $(x + h)^n$ when $n$ is a different integer?
2. What is the binomial theorem, and how can it be applied to other algebraic expressions?
3. How can you apply the expansion to find the derivative of a polynomial?
4. How does the expansion simplify when $h$ is very small?
5. What are other ways to expand binomials besides using the binomial theorem?

Tip:

When expanding binomials, knowing the coefficients (binomial coefficients) from Pascal's Triangle can make the process quicker and easier, especially for lower powers.