https://file.aiquickdraw.com/mathbot/file/17257937890487xgkiooh.jpg

The uploaded image contains several algebraic expressions that need to be simplified or expanded. I'll go through each problem step-by-step and provide the solutions.

### Problem 8: \( 2(3x + 2) \)
Expand using the distributive property:

\[
2(3x + 2) = 2 \cdot 3x + 2 \cdot 2 = 6x + 4
\]

### Problem 9: \(-2(x + 3)(x - 3)\)
Use the distributive property (FOIL) first and then multiply by \(-2\):

1. Expand \((x + 3)(x - 3)\):
   \[
   (x + 3)(x - 3) = x^2 - 9
   \]
2. Multiply by \(-2\):
   \[
   -2(x^2 - 9) = -2x^2 + 18
   \]

### Problem 10: \(-(2x - 5)^2\)
Expand the square of the binomial and multiply by \(-1\):

1. Expand \((2x - 5)^2\):
   \[
   (2x - 5)^2 = 4x^2 - 20x + 25
   \]
2. Multiply by \(-1\):
   \[
   -(4x^2 - 20x + 25) = -4x^2 + 20x - 25
   \]

### Problem 11: \((3x + 3y)(2x - 3y)\)
Expand using FOIL:

\[
(3x + 3y)(2x - 3y) = 3x \cdot 2x + 3x \cdot (-3y) + 3y \cdot 2x + 3y \cdot (-3y)
\]

Simplify:

\[
6x^2 - 9xy + 6xy - 9y^2 = 6x^2 - 3xy - 9y^2
\]

### Problem 12: \((x + 5)^2 + (x + 3)(x - 3)\)
Expand and simplify:

1. Expand \((x + 5)^2\):
   \[
   (x + 5)^2 = x^2 + 10x + 25
   \]
2. Expand \((x + 3)(x - 3)\):
   \[
   (x + 3)(x - 3) = x^2 - 9
   \]
3. Combine:
   \[
   x^2 + 10x + 25 + x^2 - 9 = 2x^2 + 10x + 16
   \]

### Problem 13: \((3x - y)^3 + 5(x + y)(x - y)\)
Simplify each part:

1. Expand \((3x - y)^3\):
   Use the binomial expansion formula: 
   \[
   (3x - y)^3 = 27x^3 - 27x^2y + 9xy^2 - y^3
   \]

2. Expand \(5(x + y)(x - y)\):
   \[
   (x + y)(x - y) = x^2 - y^2
   \]
   So, 
   \[
   5(x^2 - y^2) = 5x^2 - 5y^2
   \]

Final expression:
\[
27x^3 - 27x^2y + 9xy^2 - y^3 + 5x^2 - 5y^2
\]

### Problem 14: \(4(x - 3y)(x + 3y) - (2x - y)^2\)
Simplify each part:

1. Expand \(4(x - 3y)(x + 3y)\):
   \[
   (x - 3y)(x + 3y) = x^2 - 9y^2
   \]
   So, 
   \[
   4(x^2 - 9y^2) = 4x^2 - 36y^2
   \]

2. Expand \((2x - y)^2\):
   \[
   (2x - y)^2 = 4x^2 - 4xy + y^2
   \]

Combine:
\[
4x^2 - 36y^2 - 4x^2 + 4xy - y^2 = 4xy - 37y^2
\]

### Problem 15: \((2x + 5)(2x - 5) + 4(x - 2)^2\)
Simplify each part:

1. Expand \((2x + 5)(2x - 5)\):
   \[
   (2x + 5)(2x - 5) = 4x^2 - 25
   \]

2. Expand \(4(x - 2)^2\):
   \[
   (x - 2)^2 = x^2 - 4x + 4
   \]
   So,
   \[
   4(x^2 - 4x + 4) = 4x^2 - 16x + 16
   \]

Combine:
\[
4x^2 - 25 + 4x^2 - 16x + 16 = 8x^2 - 16x - 9
\]

If you have any questions or would like further details on any step, please let me know! 

### Follow-up Questions:
1. How can you expand other complex binomial expressions?
2. What methods can simplify polynomial expressions more efficiently?
3. Can you explain how the FOIL method works with larger polynomials?
4. How would these expressions change if one of the variables was set to zero?
5. What are some common mistakes to avoid when expanding algebraic expressions?

**Tip:** Always double-check your signs and coefficients when expanding expressions to avoid small calculation errors!

Simplify and expand the given algebraic expressions: (1) 2(3x + 2), (2) -2(x + 3)(x - 3), (3) -(2x - 5)^2, (4) (3x + 3y)(2x - 3y), (5) (x + 5)^2 + (x + 3)(x - 3), (6) (3x - y)^3 + 5(x + y)(x - y), (7) 4(x - 3y)(x + 3y) - (2x - y)^2, (8) (2x + 5)(2x - 5) + 4(x - 2)^2

This page covers detailed step-by-step solutions for simplifying and expanding complex algebraic expressions. It includes problems involving the distributive property, FOIL method, and binomial expansion. Perfect for students in Grades 8-10 who want to enhance their algebra skills.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation