Simplify and express each of the following in the form (a + ib) :
`(i)" "((5)/(-3+2i)+(2)/(1-i))((4-5i)/(3+2i))" "(ii)" "((1)/(1-4i)-(2)/(1+i))((1-i)/(5+3i))`

Explanation

To simplify the given expressions and express them in the form \( a + ib \), we will follow a systematic approach for each part of the question.

### Part (i):
Simplify and express \( \left( \frac{5}{-3 + 2i} + \frac{2}{1 - i} \right) \left( \frac{4 - 5i}{3 + 2i} \right) \).

**Step 1: Simplify \( \frac{5}{-3 + 2i} \)**
To simplify \( \frac{5}{-3 + 2i} \), we multiply the numerator and denominator by the conjugate of the denominator, which is \( -3 - 2i \):

\[ \frac{5(-3 - 2i)}{(-3 + 2i)(-3 - 2i)} = \frac{-15 - 10i}{(-3)^2 + (2)^2} = \frac{-15 - 10i}{9 + 4} = \frac{-15 - 10i}{13} \]

So,

\[ \frac{5}{-3 + 2i} = -\frac{15}{13} - \frac{10}{13}i \]

**Step 2: Simplify \( \frac{2}{1 - i} \)**
Similarly, we multiply the numerator and denominator by the conjugate of the denominator, which is \( 1 + i \):

\[ \frac{2(1 + i)}{(1 - i)(1 + i)} = \frac{2 + 2i}{1^2 + 1^2} = \frac{2 + 2i}{2} = 1 + i \]

**Step 3: Combine the two fractions**
Now we add the two results:

\[ -\frac{15}{13} - \frac{10}{13}i + 1 + i = \left(1 - \frac{15}{13}\right) + \left(1 - \frac{10}{13}\right)i \]

Calculating the real and imaginary parts:

\[ 1 - \frac{15}{13} = \frac{13}{13} - \frac{15}{13} = -\frac{2}{13} \]
\[ 1 - \frac{10}{13} = \frac{13}{13} - \frac{10}{13} = \frac{3}{13} \]

So,

\[ \frac{5}{-3 + 2i} + \frac{2}{1 - i} = -\frac{2}{13} + \frac{3}{13}i \]

**Step 4: Multiply by \( \frac{4 - 5i}{3 + 2i} \)**
Now we simplify \( \frac{4 - 5i}{3 + 2i} \) by multiplying by the conjugate of the denominator:

\[ \frac{(4 - 5i)(3 - 2i)}{(3 + 2i)(3 - 2i)} = \frac{12 - 8i - 15i + 10}{9 + 4} = \frac{22 - 23i}{13} \]

**Step 5: Multiply the results**
Now we multiply:

\[ \left(-\frac{2}{13} + \frac{3}{13}i\right) \cdot \frac{22 - 23i}{13} \]

Calculating this gives:

\[ \frac{(-2)(22) + (-2)(-23i) + (3i)(22) + (3i)(-23i)}{169} \]
\[ = \frac{-44 + 46i + 66i + 69}{169} = \frac{25 + 112i}{169} \]

Thus, the final result for part (i) is:

\[ \frac{25}{169} + \frac{112}{169}i \]

### Part (ii):
Simplify and express \( \left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right) \left( \frac{1 - i}{5 + 3i} \right) \).

**Step 1: Simplify \( \frac{1}{1 - 4i} \)**
Multiply by the conjugate:

\[ \frac{1(1 + 4i)}{(1 - 4i)(1 + 4i)} = \frac{1 + 4i}{1 + 16} = \frac{1 + 4i}{17} \]

**Step 2: Simplify \( \frac{2}{1 + i} \)**
Multiply by the conjugate:

\[ \frac{2(1 - i)}{(1 + i)(1 - i)} = \frac{2 - 2i}{1 + 1} = \frac{2 - 2i}{2} = 1 - i \]

**Step 3: Combine the two fractions**
Now we subtract:

\[ \frac{1 + 4i}{17} - (1 - i) = \frac{1 + 4i - 17 + 17i}{17} = \frac{-16 + 21i}{17} \]

**Step 4: Multiply by \( \frac{1 - i}{5 + 3i} \)**
First, simplify \( \frac{1 - i}{5 + 3i} \):

\[ \frac{(1 - i)(5 - 3i)}{(5 + 3i)(5 - 3i)} = \frac{5 - 3i - 5i - 3(-1)}{25 + 9} = \frac{8 - 8i}{34} = \frac{4 - 4i}{17} \]

**Step 5: Multiply the results**
Now multiply:

\[ \left(\frac{-16 + 21i}{17}\right) \cdot \left(\frac{4 - 4i}{17}\right) = \frac{(-16)(4) + (-16)(-4i) + (21i)(4) + (21i)(-4i)}{289} \]
\[ = \frac{-64 + 64i + 84i - 84}{289} = \frac{-148 + 148i}{289} \]

Thus, the final result for part (ii) is:

\[ -\frac{148}{289} + \frac{148}{289}i \]

### Final Answers:
- Part (i): \( \frac{25}{169} + \frac{112}{169}i \)
- Part (ii): \( -\frac{148}{289} + \frac{148}{289}i \)