Based on the image provided, the question requires describing and correcting the student's error when dividing complex numbers.

Analysis:

1. The problem is about dividing $\frac{1+i}{3-i}$.
2. The correct method to divide complex numbers involves multiplying both numerator and denominator by the complex conjugate of the denominator, $3+i$, not their respective conjugates.

The Student's Error:

The student multiplied both the numerator and denominator by their own complex conjugates instead of multiplying both by the complex conjugate of the denominator. This error is indicated by option B in the provided choices.

Correct Answer:

The correct process should involve:

1. Multiplying the numerator and denominator by $3+i$ (the complex conjugate of $3-i$). $\frac{1+i}{3-i} \cdot \frac{3+i}{3+i} = \frac{(1+i)(3+i)}{(3-i)(3+i)}$
2. Simplifying the numerator and denominator:
   • Numerator: $(1+i)(3+i) = 3 + i + 3i + i^2 = 3 + 4i - 1 = 2 + 4i$
   • Denominator: $(3-i)(3+i) = 3^2 - (-i^2) = 9 + 1 = 10$ $\text{Result: } \frac{2+4i}{10} = \frac{2}{10} + \frac{4i}{10} = \frac{1}{5} + \frac{2i}{5}.$

Final Correct Answer:

$\frac{1}{5} + \frac{2i}{5}$

Would you like further clarification on any of the steps? Here are some related questions:

1. Why is multiplying by the complex conjugate necessary in division of complex numbers?
2. What are the rules for simplifying imaginary numbers like $i^2$?
3. How would the process change if the numerator were not $1+i$ but something more complex?
4. How can you check your work when dividing complex numbers?
5. Are there other applications for the conjugate method in complex arithmetic?

Tip: Always simplify the final result of complex number calculations to standard form $a + bi$, where $a$ and $b$ are real numbers.