To solve the problem of dividing [tex]\(\left(2 + 2x^2\right) \cdot (x + 2)\)[/tex] by [tex]\(x\)[/tex], we need to follow these steps:
### Step 1: Formulate the Dividend
First, we need to expand the expression [tex]\(\left(2 + 2x^2\right) \cdot (x + 2)\)[/tex]:
[tex]\[
(x + 2) \cdot (2x^2 + 2) = (x + 2) \left(2x^2 + 2\right)
\][/tex]
Multiplying out the terms:
[tex]\[
(2x^2 + 2) \cdot (x + 2) = (2x^2 \cdot x) + (2x^2 \cdot 2) + (2 \cdot x) + (2 \cdot 2)
\][/tex]
[tex]\[
= 2x^3 + 4x^2 + 2x + 4
\][/tex]
Thus, the expanded form of the dividend is:
[tex]\[
2x^3 + 4x^2 + 2x + 4
\][/tex]
### Step 2: Formulate the Problem
We need to divide the above polynomial by [tex]\(x\)[/tex]. So the problem is:
[tex]\[
\frac{2x^3 + 4x^2 + 2x + 4}{x}
\][/tex]
### Step 3: Perform the Division
Let's divide each term of the polynomial [tex]\(2x^3 + 4x^2 + 2x + 4\)[/tex] by [tex]\(x\)[/tex]:
1. [tex]\(\frac{2x^3}{x} = 2x^2\)[/tex]
2. [tex]\(\frac{4x^2}{x} = 4x\)[/tex]
3. [tex]\(\frac{2x}{x} = 2\)[/tex]
4. [tex]\(\frac{4}{x} = \frac{4}{x}\)[/tex]
So, combining these results, we get:
[tex]\[
2x^2 + 4x + 2 + \frac{4}{x}
\][/tex]
### Step 4: Identify the Quotient and the Remainder
Here:
- Quotient is [tex]\(2x^2 + 4x + 2\)[/tex]
- Remainder is [tex]\(4\)[/tex]
### Step 5: Writing the Answer
Using the form [tex]\(\text{Dividend} = \text{Divisor} \cdot \text{Quotient} + \text{Remainder}\)[/tex], we can write the answer as:
[tex]\[
2x^3 + 4x^2 + 2x + 4 = x \cdot (2x^2 + 4x + 2) + 4
\][/tex]
Therefore, the division of [tex]\(\left(2 + 2x^2\right) \cdot (x + 2)\)[/tex] by [tex]\(x\)[/tex] results in:
Dividend: [tex]\((x + 2) \left(2x^2 + 2\right)\)[/tex]
Divisor: [tex]\(x\)[/tex]
Quotient: [tex]\(2x^2 + 4x + 2\)[/tex]
Remainder: [tex]\(4\)[/tex]
So the final form of the answer is as follows:
[tex]\[
\left(2 + 2x^2\right) \cdot (x + 2) = x \cdot (2x^2 + 4x + 2) + 4
\][/tex]
