`(2/(1-i) + 3/(1+i))((2+3i)/(4+5i))`is equal to

Options

A. ` -117/82 - 13/82 i`
B. `-117/82 + 13/82 i`
C. `117/82 - (13i)/82`
D. `117/82 + ( 13i)/82 `

Answer

To solve the expression \((2/(1-i) + 3/(1+i))((2+3i)/(4+5i))\), we will follow these steps:

### Step 1: Simplify the first part \((2/(1-i) + 3/(1+i))\)

1. Find a common denominator for the fractions:
\[ \text{LCM of } (1-i) \text{ and } (1+i) = (1-i)(1+i) = 1^2 - i^2 = 1 - (-1) = 2 \]

2. Rewrite the fractions:
\[ \frac{2}{1-i} + \frac{3}{1+i} = \frac{2(1+i) + 3(1-i)}{2} \]

3. Expand the numerator:
\[ = \frac{(2 + 2i) + (3 - 3i)}{2} = \frac{5 - i}{2} \]

### Step 2: Simplify the second part \((2+3i)/(4+5i)\)

1. Multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{2+3i}{4+5i} \cdot \frac{4-5i}{4-5i} = \frac{(2+3i)(4-5i)}{(4+5i)(4-5i)} \]

2. Calculate the denominator:
\[ (4+5i)(4-5i) = 16 - 25(-1) = 16 + 25 = 41 \]

3. Calculate the numerator:
\[ (2+3i)(4-5i) = 8 - 10i + 12i - 15i^2 = 8 + 15 + 2i = 23 + 2i \]

4. So, we have:
\[ \frac{2+3i}{4+5i} = \frac{23 + 2i}{41} \]

### Step 3: Combine the two parts

Now we need to multiply the results from Step 1 and Step 2:
\[ \left(\frac{5 - i}{2}\right) \cdot \left(\frac{23 + 2i}{41}\right) \]

1. Multiply the numerators:
\[ (5 - i)(23 + 2i) = 115 + 10i - 23i - 2i^2 = 115 - 13i + 2 = 117 - 13i \]

2. Multiply the denominators:
\[ 2 \cdot 41 = 82 \]

3. Thus, we have:
\[ \frac{117 - 13i}{82} \]

### Final Answer
The final result is:
\[ \frac{117}{82} - \frac{13}{82}i \]