To simplify the expression $4(a + b)^4$, we can apply the binomial theorem to expand $(a + b)^4$, and then multiply by 4.

Step 1: Expand $(a + b)^4$ using the binomial theorem.

The binomial expansion for $(a + b)^4$ is given by:

$(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4$

Step 2: Multiply the expanded form by 4.

Now multiply each term in the expansion by 4:

$4(a + b)^4 = 4(a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4)$

This results in:

$4(a + b)^4 = 4a^4 + 16a^3b + 24a^2b^2 + 16ab^3 + 4b^4$

Final answer:

$4(a + b)^4 = 4a^4 + 16a^3b + 24a^2b^2 + 16ab^3 + 4b^4$

Would you like to go over any step in more detail or have any further questions?

Here are some related questions you may find useful:

1. How do I expand binomials using the binomial theorem?
2. What is the binomial expansion for $(x + y)^5$?
3. How does the binomial expansion relate to Pascal’s triangle?
4. How can I simplify higher powers of binomials beyond $(a + b)^4$?
5. What are some applications of binomial expansions in calculus?

Tip: When expanding binomials, always keep track of the coefficients, which correspond to the entries in Pascal's Triangle.