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It looks like you're working on expanding binomial expressions using the binomial theorem or Pascal's triangle. Let's analyze what you've written:

### Part (a) \((1 + 2x)^5\)

This appears to be the expansion of \((1 + 2x)^5\). Using the binomial theorem, the expansion of \((a + b)^n\) is given by:

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

In this case, \(a = 1\), \(b = 2x\), and \(n = 5\).

Using the binomial coefficients for \(n = 5\) (which are \(1, 5, 10, 10, 5, 1\)):

\[
(1 + 2x)^5 = 1^5 + 5(1^4)(2x) + 10(1^3)(2x)^2 + 10(1^2)(2x)^3 + 5(1)(2x)^4 + (2x)^5
\]

Simplifying:

\[
= 1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5
\]

### Part (b) \((1 - 3x)^4\)

For this, we again use the binomial theorem with \(a = 1\), \(b = -3x\), and \(n = 4\).

The binomial coefficients for \(n = 4\) are \(1, 4, 6, 4, 1\). Expanding:

\[
(1 - 3x)^4 = 1^4 + 4(1^3)(-3x) + 6(1^2)(-3x)^2 + 4(1)(-3x)^3 + (-3x)^4
\]

Simplifying:

\[
= 1 - 12x + 54x^2 - 108x^3 + 81x^4
\]

### Part (c) \((1 + 2x)^5(1 - 3x)^4\)

You now want to multiply the expansions of \((1 + 2x)^5\) and \((1 - 3x)^4\):

\[
(1 + 2x)^5 = 1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5
\]
\[
(1 - 3x)^4 = 1 - 12x + 54x^2 - 108x^3 + 81x^4
\]

To get the full expansion, you'll need to multiply these two polynomials term by term. This results in a polynomial with degrees up to \(9\), but we can start by calculating the first few terms to illustrate:

\[
(1)(1) = 1
\]
\[
(1)(-12x) + (10x)(1) = -12x + 10x = -2x
\]
\[
(1)(54x^2) + (10x)(-12x) + (40x^2)(1) = 54x^2 - 120x^2 + 40x^2 = -26x^2
\]

And you would continue this process to obtain the full expansion.

Would you like more details on the full expansion, or explanations for specific parts? Let me know!

Here are 5 related questions that might help your understanding:
1. How does Pascal's triangle help simplify binomial expansions?
2. What is the significance of binomial coefficients in expansions?
3. How do the powers of \(x\) combine when multiplying two polynomials?
4. Why does the highest degree term in the product of \((1+2x)^5(1-3x)^4\) have degree 9?
5. How would the expansion change if the powers of the binomials were different?

**Tip:** When multiplying two polynomials, arrange them carefully and expand step-by-step to avoid missing terms!

(1 + 2x)^5 and (1 - 3x)^4 expansions using Pascal's Triangle, and multiplication of the results

Explore the binomial expansions of (1 + 2x)^5 and (1 - 3x)^4 using Pascal's Triangle, followed by their multiplication. This detailed solution covers essential algebraic concepts and the binomial theorem, suitable for high school students in Grades 10-12.

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