Find the values of x and y that satisfy the equation (1+2i)(x+yi) = 4+3i.

We are given the equation $(1+2i)(x+yi) = 4+3i$.
Multiply the terms on the left side:

Since $i^2 = -1$, substitute this into the equation:

For two complex numbers to be equal, their real parts must be equal, and their imaginary parts must be equal.
Equating real parts:

Equating imaginary parts:

From Equation 2, we can express $y$ in terms of $x$:

Substitute this expression for $y$ into Equation 1:

Add 6 to both sides:

Divide by 5:

Now substitute $x=2$ back into the expression for $y$:

So, $x=2$ and $y=-1$.

The values $x=2$ and $y=-1$ match option A.

The final answer is $\boxed{\text{A. } x=2, y=-1}$.

:

We need to find the four distinct roots of the equation $z^4 = 1$.
This means we are looking for the fourth roots of unity.

The complex number 1 can be written as $1 + 0i$.
Its modulus is $r = \sqrt{1^2 + 0^2} = 1$.
Its argument is $\theta = \arctan\left(\frac{0}{1}\right) = 0$.
In polar form, $1 = 1(\cos(0) + i\sin(0))$.
To find all roots, we use the general form of the argument: $1 = \cos(0 + 2k\pi) + i\sin(0 + 2k\pi)$, where $k$ is an integer.

The $n$-th roots of a complex number $w = r(\cos\phi + i\sin\phi)$ are given by:

For $z^4 = 1$, we have $n=4$, $r=1$, and $\phi=0$.

We need to find the roots for $k=0, 1, 2, 3$.

For $k=0$:

For $k=1$:

For $k=2$:

For $k=3$:

The four distinct roots are $1, i, -1, -i$.

The roots $1, -1, i, -i$ match option C.

The final answer is $\boxed{\text{C. } (1, -1, i, -i)}$.