https://file.aiquickdraw.com/mathbot/file/1726424949807t4tk406k.jpg

The expression from the image is \((x+2)(x+3)(x-2)(x-4)\). Let's expand it step by step.

1. First, expand the first two terms \((x+2)(x+3)\):
\[
(x+2)(x+3) = x(x+3) + 2(x+3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6
\]

2. Now expand the second two terms \((x-2)(x-4)\):
\[
(x-2)(x-4) = x(x-4) - 2(x-4) = x^2 - 4x - 2x + 8 = x^2 - 6x + 8
\]

3. Finally, multiply these two results:
\[
(x^2 + 5x + 6)(x^2 - 6x + 8)
\]
We distribute each term in the first factor across the second factor:
\[
x^2(x^2 - 6x + 8) = x^4 - 6x^3 + 8x^2
\]
\[
5x(x^2 - 6x + 8) = 5x^3 - 30x^2 + 40x
\]
\[
6(x^2 - 6x + 8) = 6x^2 - 36x + 48
\]

Now, combine like terms:
\[
x^4 + (-6x^3 + 5x^3) + (8x^2 - 30x^2 + 6x^2) + (40x - 36x) + 48
\]
\[
= x^4 - x^3 - 16x^2 + 4x + 48
\]

So the expanded form is:
\[
x^4 - x^3 - 16x^2 + 4x + 48
\]

Would you like further details on any step?

Here are 5 related questions you can try:
1. How would you factor the expanded polynomial \(x^4 - x^3 - 16x^2 + 4x + 48\)?
2. What happens if we substitute \(x = 1\) in the expanded expression?
3. How does the expansion change if we replace one of the terms with a different factor, like \((x+5)\)?
4. Can this expanded polynomial have any real roots?
5. What is the degree of the expanded polynomial, and what does it tell you about its graph?

**Tip:** Always distribute carefully when expanding products of polynomials, as small errors can propagate through all terms.

Expand the expression (x+2)(x+3)(x-2)(x-4).

This page explains how to expand the polynomial expression (x+2)(x+3)(x-2)(x-4). Step-by-step instructions cover the use of distributive property and combining like terms. Suitable for students in Grades 9-12.

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