Divide the polynomial `(4x^(2) + 4x + 5)` by (2x + 1) and write the quotient and the remainder.

Answer

To divide the polynomial \(4x^2 + 4x + 5\) by \(2x + 1\), we will use polynomial long division. Here are the steps:

### Step 1: Set up the division
We write \(4x^2 + 4x + 5\) under the long division symbol and \(2x + 1\) outside.

### Step 2: Divide the leading terms
Divide the leading term of the dividend \(4x^2\) by the leading term of the divisor \(2x\):
\[ \frac{4x^2}{2x} = 2x \]
This gives us the first term of the quotient.

### Step 3: Multiply and subtract
Now, multiply \(2x\) by the entire divisor \(2x + 1\):
\[ 2x \cdot (2x + 1) = 4x^2 + 2x \]
Now subtract this from the original polynomial:
\[ (4x^2 + 4x + 5) - (4x^2 + 2x) = (4x - 2x) + 5 = 2x + 5 \]

### Step 4: Repeat the process
Now, we take the new polynomial \(2x + 5\) and divide its leading term \(2x\) by the leading term of the divisor \(2x\):
\[ \frac{2x}{2x} = 1 \]
This gives us the next term of the quotient.

### Step 5: Multiply and subtract again
Multiply \(1\) by the divisor \(2x + 1\):
\[ 1 \cdot (2x + 1) = 2x + 1 \]
Now subtract this from \(2x + 5\):
\[ (2x + 5) - (2x + 1) = 5 - 1 = 4 \]

### Step 6: Write the final result
At this point, we cannot divide further since the remainder \(4\) is of lower degree than the divisor \(2x + 1\).

Thus, the quotient is \(2x + 1\) and the remainder is \(4\).

### Final Answer:
- **Quotient**: \(2x + 1\)
- **Remainder**: \(4\)

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