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Let's solve the given problems step by step:

### Given: 
\( f(x) = 4x^3 - 6x^2 + 2x \)

### i. Express \( f(x) \) as a product of its first derivatives.
To find the derivative of \( f(x) \):
\[
f'(x) = \frac{d}{dx}(4x^3 - 6x^2 + 2x) = 12x^2 - 12x + 2
\]

Now, let's try to factorize \( f(x) \) using common factors:
\[
f(x) = 2x(2x^2 - 3x + 1)
\]

The quadratic can be further factorized:
\[
2x(2x^2 - 3x + 1) = 2x(2x - 1)(x - 1)
\]

So, the factorized form of \( f(x) \) is:
\[
f(x) = 2x(2x - 1)(x - 1)
\]

### ii. Solve the equation \( f(x) = 0 \).
Using the factorized form:
\[
2x(2x - 1)(x - 1) = 0
\]

Setting each factor equal to zero gives:
- \( 2x = 0 \Rightarrow x = 0 \)
- \( 2x - 1 = 0 \Rightarrow x = \frac{1}{2} \)
- \( x - 1 = 0 \Rightarrow x = 1 \)

So, the solutions are:
\[
x = 0, \quad x = \frac{1}{2}, \quad x = 1
\]

### iii. Given \( h(x) = 2x^2 - 3x - 9 \). Find the values of \( x \) for which \( h(x) < 1 \).
We need to solve:
\[
2x^2 - 3x - 9 < 1
\]
\[
2x^2 - 3x - 10 < 0
\]

Let's find the roots of the quadratic equation \( 2x^2 - 3x - 10 = 0 \) using the quadratic formula:
\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-10)}}{2 \cdot 2}
\]
\[
x = \frac{3 \pm \sqrt{9 + 80}}{4}
\]
\[
x = \frac{3 \pm \sqrt{89}}{4}
\]

The approximate solutions are:
\[
x_1 = \frac{3 - \sqrt{89}}{4} \quad \text{and} \quad x_2 = \frac{3 + \sqrt{89}}{4}
\]

Since this is a parabola opening upwards (coefficient of \( x^2 \) is positive), the inequality \( 2x^2 - 3x - 10 < 0 \) holds between the roots:
\[
\frac{3 - \sqrt{89}}{4} < x < \frac{3 + \sqrt{89}}{4}
\]

Would you like a more detailed explanation of any part? Here are some related questions to explore:

1. How do you factor polynomials?
2. What is the quadratic formula and how is it used?
3. How do you determine intervals for which a quadratic inequality is satisfied?
4. What is the relationship between the derivative and the factorization of a polynomial?
5. How can you graphically represent the solution to an inequality?

**Tip:** When solving inequalities involving quadratic expressions, remember to find the roots and test intervals between them.

Δίνεται η συνάρτηση f(x) = 4x^3 - 6x^2 + 2x.

i. Να αποδώσετε τη συνάρτηση f(x) ως γινόμενο πρώτων παραγόντων.
ii. Να λύσετε την εξίσωση f(x) = 0.
iii. Δίνεται η συνάρτηση h(x) = 2x^2 - 3x - 9. Να βρείτε τις τιμές του x για τις οποίες ισχύει ότι h(x) < 1.

Learn how to factorize the cubic polynomial f(x) = 4x^3 - 6x^2 + 2x, solve the equation f(x) = 0, and solve the inequality h(x) = 2x^2 - 3x - 9 < 1. This problem covers key algebraic concepts such as polynomials, quadratic equations, and inequalities. Suitable for high school students in Grades 10-12.

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