P(x)=x³+4x²+5x+5/x+2 find the quotient of the ff using the long method 

To find the quotient of the function \( P(x) = \frac{x^3 + 4x^2 + 5x + 5}{x + 2} \) using long division, we'll follow the long division process step by step.

### Step 1: Set up the division
We're dividing \( x^3 + 4x^2 + 5x + 5 \) by \( x + 2 \).

\[
\begin{array}{r|l}
  & x^2 + 2x + 1 \\
\hline
x + 2 & x^3 + 4x^2 + 5x + 5 \\
\end{array}
\]

### Step 2: Divide the leading term
- Divide the leading term of the dividend \( x^3 \) by the leading term of the divisor \( x \):
\[
\frac{x^3}{x} = x^2
\]
- Multiply \( x^2 \) by the divisor \( x + 2 \):
\[
x^2 \cdot (x + 2) = x^3 + 2x^2
\]

### Step 3: Subtract the result from the original polynomial
- Subtract \( x^3 + 2x^2 \) from the original polynomial \( x^3 + 4x^2 + 5x + 5 \):
\[
(x^3 + 4x^2 + 5x + 5) - (x^3 + 2x^2) = 2x^2 + 5x + 5
\]

### Step 4: Repeat the process
- Divide the new leading term \( 2x^2 \) by the leading term of the divisor \( x \):
\[
\frac{2x^2}{x} = 2x
\]
- Multiply \( 2x \) by the divisor \( x + 2 \):
\[
2x \cdot (x + 2) = 2x^2 + 4x
\]
- Subtract \( 2x^2 + 4x \) from the current polynomial \( 2x^2 + 5x + 5 \):
\[
(2x^2 + 5x + 5) - (2x^2 + 4x) = x + 5
\]

### Step 5: Continue the process
- Divide the new leading term \( x \) by the leading term of the divisor \( x \):
\[
\frac{x}{x} = 1
\]
- Multiply \( 1 \) by the divisor \( x + 2 \):
\[
1 \cdot (x + 2) = x + 2
\]
- Subtract \( x + 2 \) from the current polynomial \( x + 5 \):
\[
(x + 5) - (x + 2) = 3
\]

### Step 6: Final Result
The quotient is \( x^2 + 2x + 1 \) and the remainder is \( 3 \).

Thus, 
\[
\frac{x^3 + 4x^2 + 5x + 5}{x + 2} = x^2 + 2x + 1 + \frac{3}{x+2}
\]

Would you like further details on any part of the solution?

Here are some related questions you might be interested in:
1. How do you perform synthetic division?
2. What is the Remainder Theorem?
3. How do you factor polynomials?
4. What is the relationship between division of polynomials and roots?
5. How does polynomial long division differ from numerical long division?

**Tip:** When performing polynomial division, ensure you align like terms to avoid mistakes.

P(x)=x³+4x²+5x+5/x+2 find the quotient of the ff using the long method

Learn how to find the quotient of the polynomial P(x) = x³ + 4x² + 5x + 5 divided by (x + 2) using the long division method. This problem covers polynomial division techniques, suitable for high school students.

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