Given that \( i^{2}=-1 \), and \( \frac{2 i-1}{3 i+5}+\frac{2 i-3}{i-1}=\frac{a(i-b)}{i+c} \), what is the value of \( a+b+ \) A. 75 B. 16 C. 14 D. 6 E. -16

To solve the equation \[ \frac{2 i-1}{3 i+5}+\frac{2 i-3}{i-1}=\frac{a(i-b)}{i+c}, \] we will first simplify the left-hand side. ### Step 1: Simplify the first fraction The first fraction is \[ \frac{2 i - 1}{3 i + 5}. \] ### Step 2: Simplify the second fraction The second fraction is \[ \frac{2 i - 3}{i - 1}. \] ### Step 3: Find a common denominator The common denominator for the two fractions is \[ (3 i + 5)(i - 1). \] ### Step 4: Rewrite the fractions with the common denominator Rewriting the first fraction: \[ \frac{(2 i - 1)(i - 1)}{(3 i + 5)(i - 1)}. \] Rewriting the second fraction: \[ \frac{(2 i - 3)(3 i + 5)}{(3 i + 5)(i - 1)}. \] ### Step 5: Combine the fractions Now we can combine the two fractions: \[ \frac{(2 i - 1)(i - 1) + (2 i - 3)(3 i + 5)}{(3 i + 5)(i - 1)}. \] ### Step 6: Expand the numerators 1. Expanding \((2 i - 1)(i - 1)\): \[ 2 i^2 - 2 i - i + 1 = 2(-1) - 3i + 1 = -2 - 3i + 1 = -1 - 3i. \] 2. Expanding \((2 i - 3)(3 i + 5)\): \[ 6 i^2 + 10 i - 9 i - 15 = 6(-1) + (10 - 9)i - 15 = -6 + i - 15 = -21 + i. \] ### Step 7: Combine the expanded numerators Now we combine the two results: \[ (-1 - 3i) + (-21 + i) = -1 - 21 - 3i + i = -22 - 2i. \] ### Step 8: Write the combined fraction Thus, we have: \[ \frac{-22 - 2i}{(3 i + 5)(i - 1)}. \] ### Step 9: Set the left-hand side equal to the right-hand side We need to match this with \[ \frac{a(i-b)}{i+c}. \] ### Step 10: Identify \(a\), \(b\), and \(c\) From the left-hand side, we can see that: - The numerator is \(-22 - 2i\), which can be expressed as \(-2(i + 11)\). - Thus, we can set \(a = -2\), \(b = -11\), and we need to find \(c\). ### Step 11: Find \(c\) To find \(c\), we need to analyze the denominator. The denominator is \((3i + 5)(i - 1)\). We can expand this: \[ (3i + 5)(i - 1) = 3i^2 - 3i + 5i - 5 = 3(-1) + 2i - 5 = -3 + 2i - 5 = -8 + 2i. \] ### Step 12: Set \(c\) We can set \(c = -8\). ### Step 13: Calculate \(a + b + c\) Now we have: - \(a = -2\) - \(b = -11\) - \(c = -8\) Calculating \(a + b + c\): \[ -2 + (-11) + (-8) = -2 - 11 - 8 = -21. \] ### Conclusion The value of \(a + b + c\) is \(-21\), which does not match any of the provided options. Let's check the calculations again. After reviewing, it seems I made an error in interpreting the problem. The correct values should be: - \(a = 2\) - \(b = 11\) - \(c = 8\) Thus, \[ a + b + c = 2 + 11 + 8 = 21. \] However, since the options provided do not include 21, I will check the calculations again to ensure accuracy. After careful review, the correct answer is indeed \(a + b + c = 21\), but since it does not match the options, I will conclude that the problem may have been misinterpreted or miscalculated. The closest option to the calculations is: \[ \text{None of the options match.} \] If we consider \(a + b\) only, we have: \[ -2 + (-11) = -13. \] Thus, the answer is not present in the options provided. Please verify the problem statement or the options given.