The image contains a mathematical exercise sheet in French. I'll extract and solve the problems step by step, explaining each in detail.

Exercise 1:

1. Solve in $\mathbb{R}$: $x^2 - 3x + 2 = 0$

This is a quadratic equation. Factorize the quadratic: $x^2 - 3x + 2 = (x - 1)(x - 2) = 0$ The solutions are: $x = 1 \quad \text{and} \quad x = 2$

2a. Determine the sign of $2x + 1$

The expression $2x + 1$ is linear. It changes sign at $x = -\frac{1}{2}$. The sign chart is:

• $2x + 1 > 0$ for $x > -\frac{1}{2}$
• $2x + 1 < 0$ for $x < -\frac{1}{2}$

2b. Solve $2x + 1 < 0$

From the sign chart, $2x + 1 < 0$ when: $x < -\frac{1}{2}$

3. Analyze the equation $x^2 + 8x + 15 = 0$

a. Verify $\Delta = b^2 - 4ac$: $\Delta = 8^2 - 4(1)(15) = 64 - 60 = 4$

b. Solutions of the equation: $x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-8 \pm 2}{2}$ $x_1 = -3, \quad x_2 = -5$

c. Create a sign chart for $x^2 + 8x + 15$:

• Roots: $x = -3$ and $x = -5$
• Sign chart:
  • Positive for $x < -5$ and $x > -3$
  • Negative for $-5 < x < -3$

4. Solve $x^2 + 8x + 15 < 0$:

From the sign chart: $-5 < x < -3$

5. Solve using substitution for the system: