To divide the complex numbers $\frac{3 + 4i}{2 - i}$, we'll multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $2 - i$ is $2 + i$.

Step-by-step solution:

$\frac{3 + 4i}{2 - i} \times \frac{2 + i}{2 + i}$

Multiply the numerators and denominators:

Numerator:

$(3 + 4i)(2 + i) = 3(2 + i) + 4i(2 + i)$ $= 6 + 3i + 8i + 4i^2$ Recall that $i^2 = -1$, so: $= 6 + 3i + 8i - 4$ $= 2 + 11i$

Denominator:

$(2 - i)(2 + i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5$

Final result:

$\frac{3 + 4i}{2 - i} = \frac{2 + 11i}{5} = \frac{2}{5} + \frac{11}{5}i$

So the quotient is:

$\frac{2}{5} + \frac{11}{5}i$

Would you like to go over any part of the solution or need further clarification?

Here are some related questions you might find interesting:

1. How do we multiply complex numbers in general?
2. What is the significance of the conjugate of a complex number in division?
3. Can complex numbers be divided without using the conjugate? How?
4. How do we represent division of complex numbers in polar form?
5. What happens if we divide a complex number by zero?

Tip: Always remember that multiplying by the conjugate is a handy technique when dividing complex numbers!