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To help you with the performance task, I can guide you through creating two real-life problems involving the factors of polynomials and then solve them completely. Let's break down the steps based on the requirements in the task:

### 1. **Formulate two real-life problems involving factors of polynomials:**

#### Problem 1:
**Scenario**: A rectangular garden has an area represented by the quadratic expression \(x^2 + 5x + 6\) square meters. The length of the garden is \(x + 3\) meters. What is the width of the garden?

#### Problem 2:
**Scenario**: The revenue \(R(x)\) from selling a product is represented by the polynomial \(x^2 - 6x - 16\), where \(x\) is the number of products sold. Factor this polynomial to find the two possible sales figures that result in zero revenue.

---

### 2. **Solve the problems accurately and show the complete solution:**

#### **Solution to Problem 1:**
We know that the area \(A\) of a rectangle is the product of its length and width:

\[
A = \text{Length} \times \text{Width}
\]

In this case, the area is given by the polynomial \(x^2 + 5x + 6\), and the length is \(x + 3\). So we need to find the width.

**Step 1: Factor the quadratic expression**:
\[
x^2 + 5x + 6 = (x + 3)(x + 2)
\]

**Step 2: Use the factored form**:
Now, since the area \(A = (x + 3)(x + 2)\), and we already know that the length is \(x + 3\), it follows that the width is:

\[
\text{Width} = x + 2
\]

Thus, the width of the garden is \(x + 2\) meters.

#### **Solution to Problem 2:**
We are given the revenue polynomial \(R(x) = x^2 - 6x - 16\) and need to factor it to find the values of \(x\) where revenue is zero (i.e., the break-even points).

**Step 1: Factor the quadratic expression**:
\[
x^2 - 6x - 16 = (x - 8)(x + 2)
\]

**Step 2: Set the factored form equal to zero**:
To find the break-even points, we solve for \(x\) when \(R(x) = 0\):

\[
(x - 8)(x + 2) = 0
\]

**Step 3: Solve for \(x\)**:
\[
x - 8 = 0 \quad \text{or} \quad x + 2 = 0
\]
\[
x = 8 \quad \text{or} \quad x = -2
\]

Thus, the revenue is zero when \(x = 8\) or \(x = -2\). Since negative values for sales don't make sense, the number of products that results in zero revenue is \(x = 8\).

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### Recap:
- **Problem 1**: The width of the garden is \(x + 2\) meters.
- **Problem 2**: The number of products that results in zero revenue is \(x = 8\).

Would you like more details on any of the steps, or do you have any questions?

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Here are five related questions to expand your understanding:
1. What are the steps to factor different types of quadratic expressions?
2. How can we apply factoring to solve more complex real-world problems?
3. How do you know when a quadratic expression can be factored?
4. Can polynomial factoring be applied to areas beyond mathematics?
5. How does solving factored polynomials relate to finding roots or solutions?

**Tip**: Always check the discriminant (\(b^2 - 4ac\)) before factoring quadratics to see if the polynomial can be factored easily.

Formulate two real-life problems involving factors of polynomials and solve them accurately.

Explore two real-life problems involving factors of polynomials and solve them step-by-step. This task involves factoring quadratic equations and understanding how to apply them in real-world scenarios. Suitable for Grade 8 students.

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