https://file.aiquickdraw.com/mathbot/file/17238190878884tk5s499.jpg

Let's go through the steps to solve the problems presented in the image.

### Part (a): Expand and Simplify

The given expression is:
\[ (x + 1)(x - 1)(x + 2)(x - 2) \]

This expression can be expanded in the following steps:

1. **First, expand the pairs inside the brackets**:
   \[
   (x + 1)(x - 1) = x^2 - 1
   \]
   \[
   (x + 2)(x - 2) = x^2 - 4
   \]

2. **Now, multiply these results**:
   \[
   (x^2 - 1)(x^2 - 4)
   \]

3. **Expand the multiplication**:
   \[
   x^2(x^2 - 4) - 1(x^2 - 4) = x^4 - 4x^2 - x^2 + 4 = x^4 - 5x^2 + 4
   \]

So, the expanded and simplified expression is:
\[
x^4 - 5x^2 + 4
\]

### Part (b): Identify \(g\) and \(c\)

We are given:
\[ (x - 9)(x + 7)(x + 10) \]
and need to express it in the form:
\[ x^3 + gx^2 + cx + 630 \]

Let's start expanding the expression:

1. **Expand the first two factors**:
   \[
   (x - 9)(x + 7) = x^2 - 2x - 63
   \]

2. **Multiply this result by the third factor**:
   \[
   (x^2 - 2x - 63)(x + 10) = x^3 + 10x^2 - 2x^2 - 20x - 63x - 630
   \]

3. **Simplify the expression**:
   \[
   x^3 + 8x^2 - 83x - 630
   \]

Comparing with the required form \(x^3 + gx^2 + cx + 630\), we have:
- \( g = 8 \)
- \( c = -83 \)

### Summary
- **For part (a):** The expanded and simplified expression is \(x^4 - 5x^2 + 4\).
- **For part (b):** The values of \(g\) and \(c\) are \(g = 8\) and \(c = -83\).

Would you like further details or have any other questions?

### Related Questions:
1. How can you factorize the expression \(x^4 - 5x^2 + 4\) obtained in part (a)?
2. What are the roots of the polynomial \(x^4 - 5x^2 + 4\)?
3. How can you verify that the expansion in part (a) is correct?
4. Can the expression in part (b) be factored into simpler components?
5. What is the discriminant of the quadratic factors in part (b)?
6. How would the expansion differ if the signs in part (b) were different?
7. Can we graph the polynomial in part (a) to visualize its roots?
8. What are some other methods to simplify the expression in part (a)?

### Tip:
When expanding expressions, work systematically through pairs of terms to avoid errors, and always compare your result to the required form to identify coefficients.

Explore how to expand and simplify polynomial expressions with detailed step-by-step solutions. This page covers polynomial expansion techniques, focusing on quadratic equations and higher-order polynomials. Suitable for advanced high school students and learners interested in algebraic manipulations.

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