**Part A:** The area of a square is (4x2 − 12x + 9) square units. Determine the length of each side of the square by factoring the area expression completely. Show your work. (5 points)

**Part B:** The area of a rectangle is (16x2 − 9y2) square units. Determine the dimensions of the rectangle by factoring the area expression completely. Show your work. (5 points)

**Question 3** (Essay Worth 10 points)

(07.04 MC)

Given the expression: 4x10 − 64x2

**Part A:** Rewrite the expression by factoring out the greatest common factor. (4 points)

**Part B:** Factor the entire expression completely. Show the steps of your work. (6 points)

**Question 4** (Essay Worth 10 points)

(07.03 MC)

Given the expression: 12x2 + 32x − 12

**Part A:** What is the greatest common factor? Explain how to find it. (3 points)

**Part B:** Factor the expression completely. Show all necessary steps. (5 points)

**Part C:** Check your factoring from Part B by multiplying. Show all necessary steps. (2 points)

Let's break down each part of the questions step by step.

### **Part A: Area of a Square**
The area of a square is given as \(4x^2 - 12x + 9\) square units. We need to find the length of each side by factoring the expression.

1. **Expression**: \(4x^2 - 12x + 9\)

2. **Factor the quadratic expression**:
   - We look for factors of the form \((ax + b)(cx + d)\) that multiply to give the expression.
   - First, identify the coefficients: \(a = 4\), \(b = -12\), and \(c = 9\).
   - Notice that this expression can be factored as a perfect square trinomial:
   \[
   4x^2 - 12x + 9 = (2x - 3)^2
   \]

3. **Result**: Since the area of a square is \( \text{side}^2 \), each side of the square is \( 2x - 3 \) units.

### **Part B: Area of a Rectangle**
The area of a rectangle is given as \(16x^2 - 9y^2\) square units. We need to determine the dimensions of the rectangle by factoring.

1. **Expression**: \(16x^2 - 9y^2\)

2. **Factor the expression**:
   - Recognize this as a difference of squares:
   \[
   16x^2 - 9y^2 = (4x)^2 - (3y)^2 = (4x - 3y)(4x + 3y)
   \]

3. **Result**: The dimensions of the rectangle are \(4x - 3y\) and \(4x + 3y\) units.

### **Question 3: Factoring the Expression \(4x^{10} - 64x^2\)**

#### **Part A: Factoring out the Greatest Common Factor**
1. **Expression**: \(4x^{10} - 64x^2\)
2. **Greatest Common Factor (GCF)**:
   - The GCF of the terms \(4x^{10}\) and \(64x^2\) is \(4x^2\).
   - Factor out \(4x^2\):
   \[
   4x^{10} - 64x^2 = 4x^2(x^8 - 16)
   \]

#### **Part B: Factor the Entire Expression Completely**
1. **Expression after factoring out the GCF**: \(4x^2(x^8 - 16)\)
2. **Further factor \(x^8 - 16\)**:
   - Recognize \(x^8 - 16\) as a difference of squares:
   \[
   x^8 - 16 = (x^4 - 4)(x^4 + 4)
   \]
   - Further factor \(x^4 - 4\) as:
   \[
   x^4 - 4 = (x^2 - 2)(x^2 + 2)
   \]

3. **Final Factored Form**:
   \[
   4x^2(x^2 - 2)(x^2 + 2)(x^4 + 4)
   \]

### **Question 4: Factoring \(12x^2 + 32x - 12\)**

#### **Part A: Greatest Common Factor**
1. **Expression**: \(12x^2 + 32x - 12\)
2. **Find the GCF**:
   - The GCF of \(12x^2\), \(32x\), and \(-12\) is 4.
   - Factor out the GCF:
   \[
   12x^2 + 32x - 12 = 4(3x^2 + 8x - 3)
   \]

#### **Part B: Factor the Expression Completely**
1. **Expression after factoring out the GCF**: \(4(3x^2 + 8x - 3)\)
2. **Factor the quadratic \(3x^2 + 8x - 3\)**:
   - Find two numbers that multiply to \((3)(-3) = -9\) and add to 8. These numbers are 9 and -1.
   - Factor by grouping:
   \[
   3x^2 + 8x - 3 = 3x^2 + 9x - x - 3
   \]
   - Group and factor:
   \[
   = 3x(x + 3) - 1(x + 3) = (3x - 1)(x + 3)
   \]

3. **Final Factored Form**:
   \[
   4(3x - 1)(x + 3)
   \]

#### **Part C: Check Your Factoring**
1. **Multiply the factors**: 
   \[
   4(3x - 1)(x + 3)
   \]
   - First, multiply \( (3x - 1)(x + 3) \):
   \[
   3x(x + 3) - 1(x + 3) = 3x^2 + 9x - x - 3 = 3x^2 + 8x - 3
   \]
   - Now, distribute the 4:
   \[
   4(3x^2 + 8x - 3) = 12x^2 + 32x - 12
   \]
   - This matches the original expression, confirming the factorization is correct.

### Additional Practice Questions

1. What is the square root of \(49x^2 - 36\) in factored form?
2. How do you factor \(25x^2 - 1\) completely?
3. If the area of a square is \(9x^2 + 6x + 1\), what is the length of one side?
4. How do you factor the expression \(64x^4 - 81y^4\)?
5. What is the GCF of \(20x^3 + 15x^2 - 5x\)?
6. How can \(3x^2 - 12x + 12\) be factored completely?
7. What is the factored form of \(x^4 - 81\)?
8. How do you factor the trinomial \(6x^2 - 11x - 10\)?

**Tip:** Always start by finding the greatest common factor (GCF) before attempting more complex factoring methods. This simplifies the expression and reduces potential errors.

Explore how to factor quadratic expressions to determine side lengths of squares and dimensions of rectangles. Includes step-by-step solutions for problems involving area calculations and factoring quadratic equations. Suitable for high school students studying algebra.

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