The problem in the image asks to find $z_1 + z_2$ given:

$z_1 = 2 + 3i \quad \text{and} \quad z_2 = -1 + 2i.$

Solution:

Add the two complex numbers:

$z_1 + z_2 = (2 + 3i) + (-1 + 2i)$

Group the real and imaginary parts:

$z_1 + z_2 = (2 - 1) + (3i + 2i)$

Simplify:

$z_1 + z_2 = 1 + 5i$

Final Answer:

$z_1 + z_2 = 1 + 5i$

Would you like further explanations or additional steps?

Follow-up Questions:

1. What are the rules for adding and subtracting complex numbers?
2. Can you represent the result $z_1 + z_2 = 1 + 5i$ graphically on an Argand diagram?
3. What is the magnitude of the resulting complex number $1 + 5i$?
4. How would you find $z_1 \cdot z_2$ instead of their sum?
5. How does the addition of $z_1$ and $z_2$ change geometrically in the complex plane?

Tip:

When adding complex numbers, always group the real parts and the imaginary parts separately to avoid errors.