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### Problem 6: Solve the quadratic equation \(4x^2 - 8x - 12 = 0\)

We will solve it step-by-step:

1. Simplify the quadratic equation:
   \[
   4x^2 - 8x - 12 = 0
   \]
   Divide through by 4 to simplify:
   \[
   x^2 - 2x - 3 = 0
   \]

2. Factorize \(x^2 - 2x - 3\):
   The factors of \(-3\) that add up to \(-2\) are \(-3\) and \(1\). Thus:
   \[
   (x - 3)(x + 1) = 0
   \]

3. Solve for \(x\):
   Set each factor equal to zero:
   \[
   x - 3 = 0 \quad \text{or} \quad x + 1 = 0
   \]
   \[
   x = 3 \quad \text{and} \quad x = -1
   \]

**Answer**: \(x = -1\) and \(x = 3\), which corresponds to **option B**.

---

### Problem 7: Find the value of \(a\)

We are given:
\[
\frac{2x + 6}{(x + 2)^2} = \frac{2}{x + 2} + \frac{a}{(x + 2)^2}
\]
for \(x = -2\).

#### Step-by-step solution:

1. Substitute \(x = -2\) into the left-hand side:
   \[
   \frac{2(-2) + 6}{((-2) + 2)^2} = \frac{2(4)}{(0)^2}
   \]

   Similarly for \(a=(-2)^2Let's correctly analyze the problem with precise steps.

We are given the equation:

\[
\frac{2x + 6}{(x + 2)^2} = \frac{2}{x + 2} + \frac{a}{(x + 2)^2}.
\]

We need to find \( a \), assuming \( x \neq -2 \).

---

### Step 1: Rewrite the left-hand side
Simplify \( 2x + 6 \):
\[
\frac{2x + 6}{(x + 2)^2} = \frac{2(x + 3)}{(x + 2)^2}.
\]

---

### Step 2: Match terms with the right-hand side
The right-hand side is:
\[
\frac{2}{x + 2} + \frac{a}{(x + 2)^2}.
\]

Write both terms of the right-hand side with a common denominator of \((x + 2)^2\):
\[
\frac{2}{x + 2} = \frac{2(x + 2)}{(x + 2)^2}.
\]
So:
\[
\frac{2}{x + 2} + \frac{a}{(x + 2)^2} = \frac{2(x + 2) + a}{(x + 2)^2}.
\]

---

### Step 3: Equate the numerators
Equating the numerators of the left and right sides:
\[
2(x + 3) = 2(x + 2) + a.
\]

Simplify both sides:
\[
2x + 6 = 2x + 4 + a.
\]

---

### Step 4: Solve for \( a \)
Cancel \( 2x \) on both sides:
\[
6 = 4 + a.
\]

Solve for \( a \):
\[
a = 2.
\]

---

### Final Answer:
The value of \( a \) is \( \boxed{2} \).

---

Would you like a detailed explanation of either problem or a review of specific steps?

### Related Questions:
1. How do you factorize complex quadratic equations effectively?
2. What is the importance of substituting values in algebraic identities?
3. Can you generalize the process of finding constants like \( a \) in rational expressions?
4. How does completing the square help solve quadratic equations?
5. What are common errors to watch for when solving quadratic equations?

### Tip:
When solving equations involving fractions, always rewrite terms with a common denominator to simplify comparison.

What are the solutions of the quadratic equation 4x^2 - 8x - 12 = 0, and what is the value of 'a' in a rational expression?

This page provides step-by-step solutions to a quadratic equation 4x^2 - 8x - 12 = 0 and the determination of 'a' in a rational expression involving algebraic manipulation. Perfect for students in Grades 7-10 seeking clear explanations and practice problems.

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