https://file.aiquickdraw.com/mathbot/file/1732151718928gug1k4xh.jpg

Let me extract the mathematical content from the uploaded image and solve the questions for you.

### Question 1: Simplify and identify the equivalent expression.
Expression:
\[
-3a(a + b - 5) + 4(-2a + 2b) + b(a + 3b - 7)
\]

#### Step-by-step simplification:
1. Expand each term:
   - \( -3a(a + b - 5) = -3a^2 - 3ab + 15a \)
   - \( 4(-2a + 2b) = -8a + 8b \)
   - \( b(a + 3b - 7) = ab + 3b^2 - 7b \)

2. Combine all terms:
   \[
   -3a^2 - 3ab + 15a - 8a + 8b + ab + 3b^2 - 7b
   \]

3. Group like terms:
   - For \(a^2\): \( -3a^2 \)
   - For \(ab\): \( -3ab + ab = -2ab \)
   - For \(a\): \( 15a - 8a = 7a \)
   - For \(b^2\): \( 3b^2 \)
   - For \(b\): \( 8b - 7b = b \)

Final simplified expression:
\[
-3a^2 + 3b^2 - 2ab + 7a + b
\]

**Answer:** \( D \) \( -3a^2 + 3b^2 - 2ab + 7a + b \).

---

### Question 2: Multiply the polynomials \((x + 3)(2x - 4)\).

#### Step-by-step multiplication:
1. Use the distributive property:
   \[
   (x + 3)(2x - 4) = x(2x - 4) + 3(2x - 4)
   \]

2. Expand each term:
   - \( x(2x - 4) = 2x^2 - 4x \)
   - \( 3(2x - 4) = 6x - 12 \)

3. Combine like terms:
   \[
   2x^2 - 4x + 6x - 12 = 2x^2 + 2x - 12
   \]

**Answer:**
\[
a = 2, \, b = 2, \, c = -12
\]

---

### Question 3: Simplify \( 3 + 2(x + 4)(x - 4) \).

#### Step-by-step simplification:
1. Recognize the difference of squares:
   \[
   (x + 4)(x - 4) = x^2 - 16
   \]

2. Substitute into the expression:
   \[
   3 + 2(x^2 - 16)
   \]

3. Distribute the \(2\):
   \[
   3 + 2x^2 - 32
   \]

4. Combine constants:
   \[
   2x^2 - 29
   \]

**Answer:** \( B \) \( 2x^2 - 29 \).

---

Let me know if you'd like detailed explanations for any step or have further questions! Here are five questions to expand your understanding:

1. How can the distributive property simplify polynomial multiplication?
2. What are the steps to identify like terms when combining expressions?
3. How is the difference of squares formula applied in simplifying polynomials?
4. How does grouping terms help in simplifying large algebraic expressions?
5. Can you identify and explain when to use factoring in similar problems?

**Tip:** Always double-check signs when expanding or combining terms to avoid errors!

1. An expression is shown:
-3a(a + b - 5) + 4(-2a + 2b) + b(a + 3b - 7)
Which expression is equivalent to the expression shown?
Options:
A. -11a^2 + 3b^2 - 2ab + 7a + b
B. -11a^2 + 3b^2 - 4ab + 7a + b
C. -3a^2 + 3b^2 - 2ab + 7a + b
D. -3a^2 + 3b^2 - 4ab + 7a + b (selected)

2. Multiply the polynomials (x + 3)(2x - 4). What is the product in the form ax^2 + bx + c?
Enter your answers in the boxes: a = __, b = __, c = __

3. Which expression is equivalent to 3 + 2(x + 4)(x - 4)?
Options:
A. 2x^2 - 13
B. 2x^2 - 29 (selected)
C. 2x^2 - 35
D. 5x^2 - 80

Explore algebra problems involving polynomial simplification and multiplication. This page covers expanding expressions using the distributive property and solving polynomial multiplication with detailed step-by-step solutions. Suitable for Grades 8-10 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation