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Let's break down the two parts of this question.

### Part (c): Rewrite \( f(x) = 2x^2 + 12x + 14 \) in the form \( f(x) = a(x + h)^2 + k \)
This is asking us to **complete the square**.

1. Start with:
   \[
   f(x) = 2x^2 + 12x + 14
   \]
2. Factor out 2 from the first two terms:
   \[
   f(x) = 2(x^2 + 6x) + 14
   \]
3. Complete the square inside the parentheses. Take half of 6, square it, and add/subtract it:
   \[
   x^2 + 6x = (x + 3)^2 - 9
   \]
   So:
   \[
   f(x) = 2\left((x + 3)^2 - 9\right) + 14
   \]
4. Expand and simplify:
   \[
   f(x) = 2(x + 3)^2 - 18 + 14 = 2(x + 3)^2 - 4
   \]
   Thus, the expression in completed square form is:
   \[
   f(x) = 2(x + 3)^2 - 4
   \]

The parameters are:
- \( a = 2 \), \( h = -3 \), and \( k = -4 \).

#### Turning point:
Since the function is written as \( f(x) = a(x + h)^2 + k \), the turning point is given by:
\[
\text{Turning point} = (-h, k) = (-(-3), -4) = (3, -4)
\]

---

### Part (d): Solve \( \left| \frac{x - 2}{x + 3} \right| = 4 \)

This equation involves an **absolute value**. To solve it, we break it into two cases.

#### Case 1: \( \frac{x - 2}{x + 3} = 4 \)
\[
x - 2 = 4(x + 3)
\]
\[
x - 2 = 4x + 12
\]
\[
-2 - 12 = 4x - x \Rightarrow -14 = 3x
\]
\[
x = -\frac{14}{3}
\]

#### Case 2: \( \frac{x - 2}{x + 3} = -4 \)
\[
x - 2 = -4(x + 3)
\]
\[
x - 2 = -4x - 12
\]
\[
x + 4x = -12 + 2 \Rightarrow 5x = -10
\]
\[
x = -2
\]

Thus, the solutions to the equation are:
\[
x = -\frac{14}{3} \quad \text{and} \quad x = -2
\]

---

Do you have any questions or would you like further clarification on these steps?

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Here are 5 related questions to explore:
1. How do you find the vertex of a quadratic function without completing the square?
2. What does the value of \( a \) indicate about the shape of a parabola?
3. How would the solutions change if the absolute value equation in part (d) had a different constant on the right-hand side?
4. Can a quadratic function have more than one turning point? Why or why not?
5. How do you solve inequalities involving absolute values?

**Tip:** Always double-check your solutions when dealing with absolute value equations, as extraneous solutions can arise.

Write the expression f(x) = 2x^2 + 12x + 14 in the form f(x) = a(x + h)^2 + k and hence state the turning point of f(x). Calculate the value(s) of x that are valid for the equation |(x - 2) / (x + 3)| = 4.

Learn how to rewrite the quadratic expression f(x) = 2x^2 + 12x + 14 in the form a(x + h)^2 + k, determine its turning point, and solve the absolute value equation |(x - 2) / (x + 3)| = 4. This guide covers key concepts like completing the square and absolute value properties, suitable for students in Grades 10-12.

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