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Let's go through the exercise step by step. The function \( f(x) \) is defined as:

\[
f(x) = -2(x - 1)^2 + 8
\]

### 1. **Développer \( f(x) \)** (Expand \( f(x) \))

We expand \( f(x) \) by applying the formula for squaring a binomial:

\[
f(x) = -2 \left( (x - 1)^2 \right) + 8
\]
\[
(x - 1)^2 = x^2 - 2x + 1
\]
\[
f(x) = -2(x^2 - 2x + 1) + 8
\]
\[
f(x) = -2x^2 + 4x - 2 + 8
\]
\[
f(x) = -2x^2 + 4x + 6
\]

Thus, the expanded form of \( f(x) \) is:
\[
f(x) = -2x^2 + 4x + 6
\]

---

### 2. **Justifier que pour tout réel \( x \), on a : \( f(x) = (2x+2)(-x+3) \)**

We need to verify if the given form \( f(x) = (2x+2)(-x+3) \) is valid for all \( x \). Let's expand the right-hand side:

\[
(2x+2)(-x+3) = 2x(-x+3) + 2(-x+3)
\]
\[
= 2x(-x) + 2x(3) + 2(-x) + 2(3)
\]
\[
= -2x^2 + 6x - 2x + 6
\]
\[
= -2x^2 + 4x + 6
\]

This matches the expanded form from step 1, so the given factorization is correct:
\[
f(x) = (2x+2)(-x+3)
\]

---

### 3a. **Dresser le tableau de signes de \( f(x) \)** (Sign chart of \( f(x) \))

From the factored form \( f(x) = (2x+2)(-x+3) \), we can find the roots of the function by setting each factor to zero:

\[
2x + 2 = 0 \implies x = -1
\]
\[
-x + 3 = 0 \implies x = 3
\]

Thus, \( f(x) = 0 \) at \( x = -1 \) and \( x = 3 \). Now we examine the signs of the function between and outside these points. The intervals we consider are:

- \( x < -1 \)
- \( -1 < x < 3 \)
- \( x > 3 \)

- For \( x < -1 \), both factors \( (2x+2) \) and \( (-x+3) \) are negative, so \( f(x) > 0 \).
- For \( -1 < x < 3 \), \( (2x+2) \) is positive and \( (-x+3) \) is positive, so \( f(x) < 0 \).
- For \( x > 3 \), both factors are positive, so \( f(x) > 0 \).

### Sign Chart Summary:
- \( f(x) > 0 \) for \( x < -1 \) or \( x > 3 \).
- \( f(x) < 0 \) for \( -1 < x < 3 \).

---

### 3b. **Solutions de l'inéquation \( f(x) \geq 0 \)**

Based on the sign chart, \( f(x) \geq 0 \) when:
- \( x \leq -1 \)
- \( x \geq 3 \)

So the solution to \( f(x) \geq 0 \) is:
\[
x \in ]-\infty, -1] \cup [3, +\infty[
\]

---

### 4. **Déterminer l'extremum de la fonction \( f \)**

The vertex of the parabola \( f(x) = -2(x - 1)^2 + 8 \) occurs at \( x = 1 \) (from the vertex form). To find whether it is a maximum or minimum, we recognize that the parabola opens downwards because of the negative coefficient of \( x^2 \). Therefore, \( f(x) \) has a maximum at \( x = 1 \).

The value of the maximum is:
\[
f(1) = -2(1 - 1)^2 + 8 = 8
\]

So the maximum value is \( 8 \) at \( x = 1 \).

---

### 5. **Tableau des variations de la fonction \( f \)**

Based on the results:
- \( f(x) \) decreases from \( -\infty \) to \( x = 1 \), where it reaches a maximum of 8.
- After \( x = 1 \), \( f(x) \) decreases again to \( -\infty \).

---

### 6. **Tracer la courbe représentative de la fonction**

Now, using the information from the maximum at \( x = 1 \), the roots at \( x = -1 \) and \( x = 3 \), and the overall shape (a parabola opening downwards), you can sketch the curve. The function is positive for \( x < -1 \) and \( x > 3 \), and negative between \( -1 \) and \( 3 \).

---

Would you like more details or assistance with any specific part? Here are five related questions to expand on the concepts we've covered:

1. How can you verify if a function has a maximum or minimum using the second derivative?
2. What is the importance of the vertex form of a quadratic function in determining its extremum?
3. How do the signs of the factors in a product of functions affect the overall sign of the function?
4. Can we apply this factorization method to higher-degree polynomials? If so, how?
5. How would the solution change if the parabola opened upwards instead of downwards?

**Tip**: When solving inequalities involving quadratic functions, always consider factoring and sign changes over different intervals to determine the solution.

A function f is defined on the interval ]-∞; +∞[ by: f(x) = -2(x - 1)^2 + 8. 1. Develop f(x). 2. Justify that for all real x, we have: f(x) = (2x + 2)(-x + 3). 3a. Draw the sign table of f(x). 3b. Deduce the solutions of the inequality f(x) ≥ 0. 4. Determine the extremum (maximum or minimum) of the function f and the value of x for which it is reached. 5. Draw the variation table of the function f. 6. In the orthogonal plane (O, I, J), trace the shape of the representative curve of the function f.

Learn how to analyze the quadratic function f(x) = -2(x - 1)^2 + 8, including factorization, solving inequalities, and determining extremum points. This step-by-step guide covers advanced algebra topics like sign tables and curve sketching, suitable for Grades 10-12.

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