I see that you've uploaded an image with several mathematical problems related to complex numbers and quadratic equations. I'll help solve and explain these step by step.

Here’s a breakdown of the first few problems from the image:

14. Solution of the quadratic equation $x^2 + 2 = 0$ is:

The equation can be rewritten as: $x^2 = -2$ Taking the square root of both sides: $x = \pm \sqrt{-2}$ This simplifies to: $x = \pm \sqrt{2}i$

So, the correct answer is C. $x = \pm \sqrt{2}i$.

15. The quadratic equation with solutions $x_1 = 3$ and $x_2 = -1$ is:

The quadratic equation with known solutions can be written as: $(x - x_1)(x - x_2) = 0$ Substitute $x_1 = 3$ and $x_2 = -1$: $(x - 3)(x + 1) = 0$ Expanding this gives: $x^2 - 3x + x - 3 = 0 \quad \Rightarrow \quad x^2 - 2x - 3 = 0$

So, the correct answer is C. $x^2 - 2x - 3 = 0$.

16. The result of $(3 + 2i)(1 + 4i)$ is:

Use distributive property (FOIL): $(3 + 2i)(1 + 4i) = 3(1) + 3(4i) + 2i(1) + 2i(4i)$ $= 3 + 12i + 2i + 8i^2$ Since $i^2 = -1$, this becomes: $3 + 14i - 8 = -5 + 14i$

So, the correct answer is B. $-5 + 14i$.

17. The result of $(2 + 3i) + (-4 - 5i)$ is:

Add the real and imaginary parts separately: $(2 + 3i) + (-4 - 5i) = (2 - 4) + (3i - 5i) = -2 - 2i$

So, the correct answer is C. $-2 - 2i$.

18. A number with real part 10 and imaginary part 3 is:

This is a direct representation of a complex number: $10 + 3i$

So, the correct answer is A. $10 + 3i$.

Would you like further explanations or have any specific questions on any of these problems?

Relative questions:

1. How do you solve quadratic equations with complex solutions?
2. What is the geometric interpretation of complex numbers?
3. How can complex conjugates be used in simplifying expressions?
4. Why does multiplying complex numbers involve $i^2 = -1$?
5. What are some real-life applications of complex numbers?

Tip:

When multiplying complex numbers, always remember to use the rule $i^2 = -1$ to simplify terms involving $i^2$.