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I see you've uploaded an image with several math problems. I'll go ahead and solve them one by one starting with question 8a.

### 8a. Expand \(\frac{1}{(1 + 2x)^2}\) up to the term in \(x^3\).

This is a binomial expansion problem. We can rewrite \(\frac{1}{(1 + 2x)^2}\) as \((1 + 2x)^{-2}\), then use the binomial series expansion formula:
\[
(1 + u)^n = 1 + n u + \frac{n(n-1)}{2} u^2 + \frac{n(n-1)(n-2)}{6} u^3 + \cdots
\]
where \(u = 2x\) and \(n = -2\).

Now we expand:
\[
(1 + 2x)^{-2} = 1 + (-2)(2x) + \frac{(-2)(-3)}{2}(2x)^2 + \frac{(-2)(-3)(-4)}{6}(2x)^3 + \cdots
\]

Simplifying the terms:
\[
= 1 - 4x + 12x^2 - 32x^3 + \cdots
\]

Thus, the expansion up to the term in \(x^3\) is:
\[
\frac{1}{(1 + 2x)^2} = 1 - 4x + 12x^2 - 32x^3
\]

### 8b. Find the coefficient of \(x^2\) in the expansion of \(\frac{(1 - 3x)^3}{(1 + 2x)^2}\).

We already have the expansion for \(\frac{1}{(1 + 2x)^2}\), which is \(1 - 4x + 12x^2 - 32x^3\).

Now we expand \((1 - 3x)^3\) up to the term in \(x^2\):
\[
(1 - 3x)^3 = 1 + 3(-3x) + \frac{3(3-1)}{2}(-3x)^2 = 1 - 9x + 27x^2
\]

To find the expansion of \(\frac{(1 - 3x)^3}{(1 + 2x)^2}\), we multiply the expansions of \((1 - 3x)^3\) and \(\frac{1}{(1 + 2x)^2}\) up to the term in \(x^2\):
\[
(1 - 9x + 27x^2)(1 - 4x + 12x^2) = 1 - 9x + 27x^2 - 4x + 36x^2 - 108x^3 + \cdots
\]

Combining like terms:
\[
= 1 - 13x + 63x^2 + \cdots
\]

Thus, the coefficient of \(x^2\) is **63**.

Would you like to continue with the next part or have any further questions on this? Also, here are some related practice questions you might find helpful:

1. Expand \(\frac{1}{(1 - x)^3}\) up to \(x^3\).
2. Find the coefficient of \(x^2\) in the expansion of \((1 + 4x)^3 (1 - 2x)^2\).
3. What is the general binomial expansion of \((1 + ax)^n\)?
4. Estimate \(\frac{1}{(1 + 0.1)^2}\) using a binomial expansion.
5. Determine for what values of \(x\) the expansion \((1 - 2x)^{-2}\) converges.

**Tip**: When expanding binomials, pay attention to negative powers and coefficients for accurate results.

Expand (1 / (1 + 2x)^2) up to the term in x^3 and find the coefficient of x^2 in the expansion of ((1 - 3x)^3 / (1 + 2x)^2).

Solve the expansion of (1 / (1 + 2x)^2) up to x^3 and find the coefficient of x^2 in ((1 - 3x)^3 / (1 + 2x)^2). Learn key binomial expansion concepts and polynomial multiplication techniques, suitable for Grades 11-12.

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