The image you uploaded shows the binomial expansion of the expression $(4x - 3)^5$, and we are asked to expand it using the Binomial Theorem.

The Binomial Theorem formula for $(a + b)^n$ is:

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

In this case, $a = 4x$, $b = -3$, and $n = 5$. We can use this to expand $(4x - 3)^5$ term by term.

The general term in the expansion is:

$\binom{5}{k} (4x)^{5-k} (-3)^k$

Let’s calculate the terms and verify the expansion. math

Constants

a = 4 # for 4x b = -3 n = 5

Function to calculate the binomial expansion terms

def binomial_expansion_term(k): coefficient = math.comb(n, k) * (a ** (n - k)) * (b ** k) power_of_x = n - k return coefficient, power_of_x

Calculate all terms in the expansion of (4x - 3)^5

expansion_terms = [binomial_expansion_term(k) for k in range(n+1)] expansion_termsThe binomial expansion of $(4x - 3)^5$ results in the following terms:

$1024x^5 - 3840x^4 + 5760x^3 - 4320x^2 + 1620x - 243$

So, the full expanded expression is:

$(4x - 3)^5 = 1024x^5 - 3840x^4 + 5760x^3 - 4320x^2 + 1620x - 243$

It looks like the terms you provided (1024x^5, 5760x^3, 1620x) match part of the expansion, but some terms are missing or incorrectly placed.

Would you like more details on how each term is derived, or do you have any specific questions? Here's a set of related questions for further exploration:

1. How do binomial coefficients relate to the expansion process?
2. What does each term represent in the binomial expansion?
3. How would the result change if the power $n$ was increased?
4. What happens if $a$ or $b$ is negative in the binomial expansion?
5. How do you handle terms with larger powers (like $x^6$ or $x^7$)?

Tip: The sign of each term alternates when using the binomial expansion formula with a negative term in the binomial. Keep track of this!