explain---**Question Stem:** Which of the following complex numbers is equivalent to the expression below? **Mathematical Expression:** (3 - 5i) / (8 + 2i) **Other Relevant Text:** (Note: i = sqrt(-1)) **Options:** No options (A, B, C, D) are provided in the image.

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We need to find the equivalent form of the expression three minus five i divided by eight plus two i. To divide complex numbers, we use a special technique called rationalizing the denominator using the complex conjugate. Remember that i equals the square root of negative one, and complex numbers are written in the form a plus b i, where a is the real part and b is the imaginary part. The complex conjugate is a key concept for dividing complex numbers. For any complex number a plus b i, its conjugate is a minus b i. When we multiply a complex number by its conjugate, we get a real number: a squared plus b squared. For our denominator eight plus two i, the conjugate is eight minus two i. Multiplying them gives us sixty-eight, which is a real number. Now we apply the conjugate method by multiplying both the numerator and denominator by eight minus two i. This is equivalent to multiplying by one, so it doesn't change the value of our expression. The result is the fraction with numerator three minus five i times eight minus two i, and denominator eight plus two i times eight minus two i. Now we expand the denominator using the difference of squares pattern. Eight plus two i times eight minus two i equals eight squared minus two i squared. This gives us sixty-four minus four i squared. Since i squared equals negative one, we get sixty-four minus four times negative one, which equals sixty-four plus four, or sixty-eight. The denominator is now a real number, which was our goal. Now we expand the numerator using the FOIL method. First: three times eight equals twenty-four. Outer: three times negative two i equals negative six i. Inner: negative five i times eight equals negative forty i. Last: negative five i times negative two i equals ten i squared. Combining terms: twenty-four minus six i minus forty i plus ten i squared. Since i squared equals negative one, we get twenty-four minus forty-six i plus ten times negative one, which simplifies to fourteen minus forty-six i.