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find the quotient using the long division 

To find the quotient of the polynomial division \(\frac{x^4 + 3x^2 + 2x + 1}{x + 2}\) using long division, we proceed as follows:

### Step 1: Set up the division
The divisor is \(x + 2\), and the dividend is \(x^4 + 3x^2 + 2x + 1\).

### Step 2: Divide the first term
Divide the first term of the dividend by the first term of the divisor:

\[
\frac{x^4}{x} = x^3
\]

This is the first term of the quotient.

### Step 3: Multiply and subtract
Multiply \(x^3\) by the divisor \(x + 2\):

\[
x^3(x + 2) = x^4 + 2x^3
\]

Now, subtract this from the original dividend:

\[
(x^4 + 3x^2 + 2x + 1) - (x^4 + 2x^3) = -2x^3 + 3x^2 + 2x + 1
\]

### Step 4: Repeat the process
Divide the first term of the new dividend by the first term of the divisor:

\[
\frac{-2x^3}{x} = -2x^2
\]

This is the next term of the quotient.

Multiply \(-2x^2\) by the divisor \(x + 2\):

\[
-2x^2(x + 2) = -2x^3 - 4x^2
\]

Subtract this from the current dividend:

\[
(-2x^3 + 3x^2 + 2x + 1) - (-2x^3 - 4x^2) = 7x^2 + 2x + 1
\]

### Step 5: Continue dividing
Next, divide the first term of the new dividend by the first term of the divisor:

\[
\frac{7x^2}{x} = 7x
\]

This is the next term of the quotient.

Multiply \(7x\) by the divisor \(x + 2\):

\[
7x(x + 2) = 7x^2 + 14x
\]

Subtract this from the current dividend:

\[
(7x^2 + 2x + 1) - (7x^2 + 14x) = -12x + 1
\]

### Step 6: Final division
Divide the first term of the new dividend by the first term of the divisor:

\[
\frac{-12x}{x} = -12
\]

This is the final term of the quotient.

Multiply \(-12\) by the divisor \(x + 2\):

\[
-12(x + 2) = -12x - 24
\]

Subtract this from the current dividend:

\[
(-12x + 1) - (-12x - 24) = 25
\]

### Step 7: Conclusion
The quotient is:

\[
x^3 - 2x^2 + 7x - 12
\]

With a remainder of \(25\).

### Final Answer
\[
\text{Quotient} = x^3 - 2x^2 + 7x - 12, \quad \text{Remainder} = 25
\]

Would you like any further details or have any questions?

Here are 5 related questions for further practice:
1. How would you verify the quotient and remainder found in this division?
2. Can you perform synthetic division on the same problem and compare the results?
3. What happens if the divisor is a quadratic polynomial instead of a linear polynomial?
4. How would you handle division if the leading coefficient of the divisor is not 1?
5. How does the remainder theorem apply to this problem?

**Tip:** Always verify your division by multiplying the quotient by the divisor and adding the remainder to ensure it equals the original polynomial.

Learn how to find the quotient of a polynomial division using long division method with step-by-step instructions. This problem involves dividing a polynomial by a linear divisor and determining the quotient and remainder. Suitable for high school students.

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