((4+5i)+2i^3)/(2+i)^2 . find its a+ib form?

To find the expression \(\frac{(4 + 5i) + 2i^3}{(2 + i)^2}\) in the form \(a + ib\), let's break it down step by step.

### Step 1: Simplify the numerator
The numerator is \((4 + 5i) + 2i^3\).

Recall that \(i^2 = -1\) and \(i^3 = i \times i^2 = i \times (-1) = -i\).

So, \(2i^3 = 2 \times (-i) = -2i\).

Now, the numerator becomes:
\[
(4 + 5i) + (-2i) = 4 + 3i
\]

### Step 2: Expand the denominator
The denominator is \((2 + i)^2\).

First, expand \((2 + i)^2\) using the formula \((a + b)^2 = a^2 + 2ab + b^2\):

\[
(2 + i)^2 = 2^2 + 2 \times 2 \times i + i^2
\]

Simplify this:

\[
2^2 = 4,\quad 2 \times 2 \times i = 4i,\quad i^2 = -1
\]

So, the denominator becomes:
\[
(2 + i)^2 = 4 + 4i - 1 = 3 + 4i
\]

### Step 3: Perform the division
We need to compute \(\frac{4 + 3i}{3 + 4i}\).

To divide complex numbers, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(3 + 4i\) is \(3 - 4i\).

So, we have:
\[
\frac{4 + 3i}{3 + 4i} \times \frac{3 - 4i}{3 - 4i} = \frac{(4 + 3i)(3 - 4i)}{(3 + 4i)(3 - 4i)}
\]

### Step 4: Simplify the expression

**Denominator:**
\[
(3 + 4i)(3 - 4i) = 3^2 - (4i)^2 = 9 - 16(-1) = 9 + 16 = 25
\]

**Numerator:**
\[
(4 + 3i)(3 - 4i) = 4 \times 3 + 4 \times (-4i) + 3i \times 3 + 3i \times (-4i)
\]
\[
= 12 - 16i + 9i - 12i^2
\]
\[
= 12 - 16i + 9i + 12
\]
Since \(i^2 = -1\):
\[
= 24 - 7i
\]

### Final expression
Now, the expression becomes:
\[
\frac{24 - 7i}{25} = \frac{24}{25} - \frac{7}{25}i
\]

So, in the form \(a + ib\), the result is:
\[
\boxed{\frac{24}{25} - \frac{7}{25}i}
\]

Would you like further details on any step or have any questions?

### Follow-up questions:

1. How do you multiply two complex numbers?
2. What is the geometric interpretation of multiplying a complex number by its conjugate?
3. How does the division of complex numbers relate to polar form?
4. Can you generalize the process for dividing any two complex numbers?
5. How does the formula for the square of a binomial apply to complex numbers?

**Tip:** When simplifying expressions involving imaginary numbers, always substitute \(i^2 = -1\) to make calculations easier.

Solve the complex expression ((4 + 5i) + 2i^3)/(2 + i)^2 and convert it into the form a + ib. This solution breaks down each step, covering key algebraic concepts such as expanding binomials, simplifying imaginary numbers, and using conjugates for division.

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