The provided image contains a quadratic equation and asks to determine the type of solutions based on the discriminant. The equation is:

$2x^2 + 3x + 2 = 0$

Let's solve this step by step.

1. General form of a quadratic equation:

The general form is:

$ax^2 + bx + c = 0$

In this case:

• $a = 2$
• $b = 3$
• $c = 2$

2. Discriminant ($\Delta$):

The discriminant formula is:

$\Delta = b^2 - 4ac$

Substituting the values of $a$, $b$, and $c$:

$\Delta = 3^2 - 4(2)(2) = 9 - 16 = -7$

3. Analyzing the discriminant:

• If $\Delta > 0$, the equation has two distinct real roots.
• If $\Delta = 0$, the equation has one real repeated root.
• If $\Delta < 0$, the equation has two distinct complex roots.

Since $\Delta = -7$, which is less than 0, the equation has two distinct complex (imaginary) solutions.

Thus, the correct answer is:

$\boxed{3} \, \text{(異なる2つの虚数解をもつ)} \, \text{Two distinct complex solutions}.$

Would you like more details or have any other questions?

Here are five related questions:

1. What are the exact complex roots of the equation $2x^2 + 3x + 2 = 0$?
2. How can the quadratic formula be applied to find roots when the discriminant is negative?
3. How does the discriminant affect the type of solutions for other quadratic equations?
4. What are the geometric interpretations of complex roots on the complex plane?
5. Can you explain how to solve quadratic equations graphically?

Tip: For quadratic equations, always check the discriminant first to quickly determine the nature of the solutions!