directions: simplify each expression.
7. $5w(-7w + 3) - 2w(12 - 9w)$
8. $6x(2x - 3) - 5(2x^2 + 9x - 3)$
9. $y(y + 4) - y(y - 3) - 9y$
10. $(x - 10)(x - 4)$
11. $(n - 1)(5n - 4)$
12. $(3w + 1)(3w - 1)$
13. $(x + 4)(x^2 + 3x - 6)$
14. $(2v + 3)(4v^2 - 3v - 6)$

Question

Accepted Answer

Let's solve each problem one by one:

### Problem 7: $ 5w(-7w + 3) - 2w(12 - 9w) $
# Explanation:
## Step 1: Distribute the terms
First, distribute $ 5w $ into $ (-7w + 3) $ and $ -2w $ into $ (12 - 9w) $:
$ 5w(-7w) + 5w(3) - 2w(12) + 2w(9w) $
$ = -35w^2 + 15w - 24w + 18w^2 $

## Step 2: Combine like terms
Combine the $ w^2 $ terms and the $ w $ terms:
$ (-35w^2 + 18w^2) + (15w - 24w) $
$ = -17w^2 - 9w $

# Answer: $ -17w^2 - 9w $

### Problem 8: $ 6x(2x - 3) - 5(2x^2 + 9x - 3) $
# Explanation:
## Step 1: Distribute the terms
Distribute $ 6x $ into $ (2x - 3) $ and $ -5 $ into $ (2x^2 + 9x - 3) $:
$ 6x(2x) - 6x(3) - 5(2x^2) - 5(9x) + 5(3) $
$ = 12x^2 - 18x - 10x^2 - 45x + 15 $

## Step 2: Combine like terms
Combine the $ x^2 $ terms, $ x $ terms, and constants:
$ (12x^2 - 10x^2) + (-18x - 45x) + 15 $
$ = 2x^2 - 63x + 15 $

# Answer: $ 2x^2 - 63x + 15 $

### Problem 9: $ y(y + 4) - y(y - 3) - 9y $
# Explanation:
## Step 1: Distribute the terms
Distribute $ y $ into $ (y + 4) $ and $ -y $ into $ (y - 3) $:
$ y(y) + y(4) - y(y) + y(3) - 9y $
$ = y^2 + 4y - y^2 + 3y - 9y $

## Step 2: Combine like terms
Combine the $ y^2 $ terms and the $ y $ terms:
$ (y^2 - y^2) + (4y + 3y - 9y) $
$ = 0y^2 - 2y $
$ = -2y $

# Answer: $ -2y $

### Problem 10: $ (x - 10)(x - 4) $
# Explanation:
## Step 1: Use the distributive property (FOIL method)
Multiply the First terms, Outer terms, Inner terms, and Last terms:
- First: $ x \cdot x = x^2 $
- Outer: $ x \cdot (-4) = -4x $
- Inner: $ -10 \cdot x = -10x $
- Last: $ -10 \cdot (-4) = 40 $

## Step 2: Combine like terms
Combine the $ x $ terms:
$ x^2 - 4x - 10x + 40 $
$ = x^2 - 14x + 40 $

# Answer: $ x^2 - 14x + 40 $

### Problem 11: $ (n - 1)(5n - 4) $
# Explanation:
## Step 1: Use the distributive property (FOIL method)
Multiply the First terms, Outer terms, Inner terms, and Last terms:
- First: $ n \cdot 5n = 5n^2 $
- Outer: $ n \cdot (-4) = -4n $
- Inner: $ -1 \cdot 5n = -5n $
- Last: $ -1 \cdot (-4) = 4 $

## Step 2: Combine like terms
Combine the $ n $ terms:
$ 5n^2 - 4n - 5n + 4 $
$ = 5n^2 - 9n + 4 $

# Answer: $ 5n^2 - 9n + 4 $

### Problem 12: $ (3w + 1)(3w - 1) $
# Explanation:
## Step 1: Recognize the difference of squares
This is a difference of squares: $ (a + b)(a - b) = a^2 - b^2 $, where $ a = 3w $ and $ b = 1 $.

## Step 2: Apply the formula
$ (3w)^2 - (1)^2 $
$ = 9w^2 - 1 $

# Answer: $ 9w^2 - 1 $

### Problem 13: $ (x + 4)(x^2 + 3x - 6) $
# Explanation:
## Step 1: Distribute $ x $ and $ 4 $ into the trinomial
Distribute $ x $ into $ (x^2 + 3x - 6) $ and $ 4 $ into $ (x^2 + 3x - 6) $:
$ x(x^2) + x(3x) + x(-6) + 4(x^2) + 4(3x) + 4(-6) $
$ = x^3 + 3x^2 - 6x + 4x^2 + 12x - 24 $

## Step 2: Combine like terms
Combine the $ x^2 $ terms, $ x $ terms, and constants:
$ x^3 + (3x^2 + 4x^2) + (-6x + 12x) - 24 $
$ = x^3 + 7x^2 + 6x - 24 $

# Answer: $ x^3 + 7x^2 + 6x - 24 $

### Problem 14: $ (2v + 3)(4v^2 - 3v - 6) $
# Explanation:
## Step 1: Distribute $ 2v $ and $ 3 $ into the trinomial
Distribute $ 2v $ into $ (4v^2 - 3v - 6) $ and $ 3 $ into $ (4v^2 - 3v - 6) $:
$ 2v(4v^2) + 2v(-3v) + 2v(-6) + 3(4v^2) + 3(-3v) + 3(-6) $
$ = 8v^3 - 6v^2 - 12v + 12v^2 - 9v - 18 $

## Step 2: Combine like terms
Combine the $ v^2 $ terms, $ v $ terms, and constants:
$ 8v^3 + (-6v^2 + 12v^2) + (-12v - 9v) - 18 $
$ = 8v^3 + 6v^2 - 21v - 18 $

# Answer: $ 8v^3 + 6v^2 - 21v - 18 $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 7: \( 5w(-7w + 3) - 2w(12 - 9w) \)

Step 1: Distribute the terms

Step 2: Combine like terms

Step 1: Distribute the terms

Step 2: Combine like terms

Step 1: Distribute the terms

Step 2: Combine like terms

Answer:

Problem 8: \( 6x(2x - 3) - 5(2x^2 + 9x - 3) \)