https://r2.erweima.ai/mathbot/image/1726345160037gkci8wfw.jpg

Let's solve the problems shown in the image.

### First Problem:

The expression given is:

\[
[(x - 4) + x^2] \cdot [(x - 4) - x^2]
\]

This is a difference of squares. The formula for difference of squares is:

\[
(a + b)(a - b) = a^2 - b^2
\]

In this case:
- \( a = (x - 4) \)
- \( b = x^2 \)

Applying the difference of squares formula:

\[
[(x - 4) + x^2] \cdot [(x - 4) - x^2] = (x - 4)^2 - (x^2)^2
\]

Now expand both terms:

\[
(x - 4)^2 = x^2 - 8x + 16
\]
\[
(x^2)^2 = x^4
\]

So the final simplified expression is:

\[
x^2 - 8x + 16 - x^4
\]
Which simplifies to:
\[
- x^4 + x^2 - 8x + 16
\]

### Second Problem:

The given expression is:

\[
64a^2 - 25
\]

This is also a difference of squares. The formula is:

\[
A^2 - B^2 = (A + B)(A - B)
\]

In this case:
- \( A = 8a \)
- \( B = 5 \)

So applying the formula:

\[
64a^2 - 25 = (8a + 5)(8a - 5)
\]

### Third Problem:

The expression given is:

\[
8x^3 + y^3
\]

This is a sum of cubes. The formula for factoring a sum of cubes is:

\[
A^3 + B^3 = (A + B)(A^2 - AB + B^2)
\]

In this case:
- \( A = 2x \)
- \( B = y \)

So applying the sum of cubes formula:

\[
8x^3 + y^3 = (2x + y)\left( (2x)^2 - (2x)(y) + y^2 \right)
\]
\[
= (2x + y)(4x^2 - 2xy + y^2)
\]

Let me know if you want further clarification on any of these problems!

Here are five related questions for further practice:
1. How would you factor the expression \( (x^4 - 16) \)?
2. What happens if the sum of cubes is negative, like \( 27x^3 - 8 \)?
3. Can you explain why \( x^2 - 9 \) is a difference of squares?
4. How would you simplify \( (x + 2)^2 - (x - 2)^2 \)?
5. Factor \( 9a^2 - 49 \) using the difference of squares.

**Tip:** Always look for common patterns such as difference of squares, sum/difference of cubes, or perfect square trinomials to simplify complex expressions!

Perform the indicated operations and simplify: [(x - 4) + x^2][(x - 4) - x^2], Factor the expression 64a^2 - 25 using a special factoring formula, and Factor the expression 8x^3 + y^3 using a special factoring formula.

Learn how to simplify and factor algebraic expressions using special factoring formulas. This guide covers the difference of squares for expressions like [(x - 4) + x^2][(x - 4) - x^2], the difference of squares formula to factor 64a^2 - 25, and the sum of cubes formula for 8x^3 + y^3. Suitable for students in Grades 9-11.

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