Okay, let's see. The problem is about William Sydney Porter making a mistake in his calculation of a complex fraction. He was supposed to compute (-3 + 4i)/(1 + 2i), but instead, he forgot the minus sign and calculated (3 + 4i)/(1 + 2i), getting 11/5 - 2/5i. Now, we need to find out what he should have gotten.

First, I remember that when dealing with complex numbers in division, we usually multiply the numerator and the denominator by the conjugate of the denominator to rationalize it. The conjugate of a complex number a + bi is a - bi. So, for the denominator 1 + 2i, the conjugate would be 1 - 2i. That makes sense because multiplying (1 + 2i)(1 - 2i) gives 1 - (2i)^2, which is 1 - (-4) = 5. So that helps eliminate the imaginary part in the denominator.

So, applying that method to both the correct problem and the incorrect one he did. Let me write down both.

First, the incorrect calculation he did: (3 + 4i)/(1 + 2i). To rationalize the denominator, multiply numerator and denominator by (1 - 2i):

Numerator: (3 + 4i)(1 - 2i)
Denominator: (1 + 2i)(1 - 2i) = 1 - (2i)^2 = 1 - (-4) = 5.

So, let's compute the numerator:

3*1 = 3
3*(-2i) = -6i
4i*1 = 4i
4i*(-2i) = -8i^2

Combine these terms:
3 - 6i + 4i -8i^2

Combine like terms:
3 - 2i -8i^2

But i^2 is -1, so -8i^2 becomes +8. So:
3 - 2i +8 = 11 - 2i

Therefore, the numerator is 11 - 2i, and the denominator is 5, so the result is (11 - 2i)/5 = 11/5 - 2/5i, which matches the given answer. So that's correct.

Now, the correct problem is (-3 + 4i)/(1 + 2i). Let's do the same process.

Multiply numerator and denominator by (1 - 2i):

Numerator: (-3 + 4i)(1 - 2i)
Denominator: (1 + 2i)(1 - 2i) = 5, as before.

Let's compute the numerator:

-3*1 = -3
-3*(-2i) = 6i
4i*1 = 4i
4i*(-2i) = -8i^2

Combine terms:
-3 + 6i + 4i -8i^2

Combine like terms:
-3 + 10i -8i^2

Again, i^2 is -1, so -8i^2 becomes +8. Therefore:
-3 + 10i +8 = 5 + 10i

So the numerator is 5 + 10i, and the denominator is 5, so the result is (5 + 10i)/5 = 1 + 2i.

Wait, that seems straightforward. So the answer should be 1 + 2i. But let me verify to make sure I didn't make any mistakes in the multiplication.

Let me redo the numerator for the correct problem:

(-3 + 4i)(1 - 2i)

First term: -3*1 = -3

Second term: -3*(-2i) = +6i

Third term: 4i*1 = +4i

Fourth term: 4i*(-2i) = -8i^2 = +8 (since i^2 = -1)

Adding all terms: -3 +6i +4i +8 = (-3 +8) + (6i +4i) = 5 +10i. Yes, that's correct.

Divide by 5: 5/5 +10i/5 =1 +2i. So that's the correct answer.

But wait, let me check if there's another way to approach this problem. Maybe by recognizing that the correct answer should be the incorrect answer plus something. Since the only difference between the two problems is the sign of the real part in the numerator. So if the incorrect calculation was (3 +4i)/(1 +2i)=11/5 -2/5i, then the correct one is (-3 +4i)/(1 +2i). Let's see if we can relate the two.

Let me denote the incorrect result as A =11/5 -2/5i. Then the correct result should be -A + something? Wait, maybe not. Let me think.

Alternatively, perhaps the correct answer is ( -3 +4i )/(1 +2i) = - (3 -4i)/(1 +2i). But that might not help directly. Let me compute (3 -4i)/(1 +2i) first.

Wait, maybe not. Let's see. Let me compute (3 -4i)/(1 +2i). Multiply numerator and denominator by (1 -2i):

Numerator: (3 -4i)(1 -2i)

3*1=3

3*(-2i)=-6i

-4i*1=-4i

-4i*(-2i)=8i^2= -8

So total: 3 -6i -4i -8 = (3 -8) + (-6i -4i) = -5 -10i

Denominator:5

So (3 -4i)/(1 +2i)= (-5 -10i)/5= -1 -2i.

Therefore, ( -3 +4i )/(1 +2i)= - (3 -4i)/(1 +2i)= -(-1 -2i)=1 +2i. Which matches the previous result. So that's another way to see it. So either way, the correct answer is 1 +2i.

Therefore, the answer William should have obtained is 1 +2i.
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