Let's solve each part of the problem step by step.

Part A: Find the values of $x$ and $y$

Given: $x(1 + i)^2 + y(2 - i)^2 = 3 + 10i$ where $x$ and $y$ are real numbers.

1. Expand the complex squares:

   • $(1 + i)^2 = 1^2 + 2i + i^2 = 1 + 2i - 1 = 2i$
   • $(2 - i)^2 = 2^2 - 2 \times 2 \times i + i^2 = 4 - 4i - 1 = 3 - 4i$

   Thus, the equation becomes: $x(2i) + y(3 - 4i) = 3 + 10i$

2. Distribute $x$ and $y$: $2xi + 3y - 4yi = 3 + 10i$

   Group the real and imaginary parts: $(3y) + (2x - 4y)i = 3 + 10i$

3. Set up the system of equations:

   • For the real part: $3y = 3$
   • For the imaginary part: $2x - 4y = 10$

4. Solve the system of equations:

   From $3y = 3$, we get: $y = 1$ Substitute $y = 1$ into $2x - 4y = 10$: $2x - 4(1) = 10$ $2x - 4 = 10$ $2x = 14 \implies x = 7$

So, the values are: $x = 7, \quad y = 1$

Part B: Given $z = -3 + 4i$ and $zw = -14 + 2i$

1. Find $w$ in the form $a + ib$:

   Given $zw = -14 + 2i$ and $z = -3 + 4i$, find $w$.

   Let $w = a + ib$. Then: $(-3 + 4i)(a + ib) = -14 + 2i$

   Expand and equate real and imaginary parts: $-3a - 3ib + 4ai + 4ib^2 = -14 + 2i$ $(-3a - 4b) + (4a - 3b)i = -14 + 2i$

   Set up the system of equations: $-3a - 4b = -14 \quad \text{(Real part)}$ $4a - 3b = 2 \quad \text{(Imaginary part)}$

   Solve these equations:

   From $-3a - 4b = -14$: $3a + 4b = 14$

   Solve the system:

   3a + 4b = 14 \\ 4a - 3b = 2 \end{cases}$$ Multiply the first equation by 3 and the second by 4: $$9a + 12b = 42$$ $$16a - 12b = 8$$ Add the equations: $$25a = 50 \implies a = 2$$ Substitute $$a = 2$$ into $$3a + 4b = 14$$: $$3(2) + 4b = 14 \implies 6 + 4b = 14 \implies 4b = 8 \implies b = 2$$ Therefore, $$w = 2 + 2i$$.

2. Find the modulus and argument of $w$:

   • Modulus: $|w| = \sqrt{a^2 + b^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$

   • Argument: $\arg(w) = \tan^{-1}\left(\frac{b}{a}\right) = \tan^{-1}\left(\frac{2}{2}\right) = \tan^{-1}(1) = \frac{\pi}{4}$

Summary:

1. Values of $x$ and $y$: $x = 7, \quad y = 1$

2. For $w$:

   • $w = 2 + 2i$
   • $|w| = 2\sqrt{2}$
   • $\arg(w) = \frac{\pi}{4}$

Do you want details or have any questions?

Here are 5 relative questions for you:

1. How do you multiply two complex numbers?
2. How is the modulus of a complex number calculated?
3. What are the standard forms of complex numbers?
4. How does the argument of a complex number relate to its position on the complex plane?
5. What does solving simultaneous equations involve?

Tip: When solving for real and imaginary parts in complex equations, separate the components carefully to avoid errors.