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Help me with this please and put in in the form of long devision

To solve the problem and express it in the form of long division, let's break it down step by step.

### Problem Statement:
You are given the equation:
\[
\frac{2x^3 + 3x^2 - 7x + 5}{x + 3} = ax^2 + bx + c + \frac{d}{x + 3}
\]
and you need to find the coefficients \( a \), \( b \), \( c \), and \( d \).

### Step 1: Perform Polynomial Long Division

Let's divide the polynomial \( 2x^3 + 3x^2 - 7x + 5 \) by \( x + 3 \) using long division.

#### 1. Divide the leading term of the dividend by the leading term of the divisor:
\[
\frac{2x^3}{x} = 2x^2
\]
So, the first term of the quotient is \( 2x^2 \).

#### 2. Multiply the entire divisor by \( 2x^2 \):
\[
2x^2(x + 3) = 2x^3 + 6x^2
\]
Subtract this from the original polynomial:
\[
(2x^3 + 3x^2 - 7x + 5) - (2x^3 + 6x^2) = -3x^2 - 7x + 5
\]

#### 3. Divide the new leading term by the leading term of the divisor:
\[
\frac{-3x^2}{x} = -3x
\]
So, the next term in the quotient is \( -3x \).

#### 4. Multiply the entire divisor by \( -3x \):
\[
-3x(x + 3) = -3x^2 - 9x
\]
Subtract this from the current polynomial:
\[
(-3x^2 - 7x + 5) - (-3x^2 - 9x) = 2x + 5
\]

#### 5. Divide the new leading term by the leading term of the divisor:
\[
\frac{2x}{x} = 2
\]
So, the next term in the quotient is \( 2 \).

#### 6. Multiply the entire divisor by \( 2 \):
\[
2(x + 3) = 2x + 6
\]
Subtract this from the current polynomial:
\[
(2x + 5) - (2x + 6) = -1
\]

### Step 2: Express the Result
Now, combining all the terms from the division, we get:
\[
\frac{2x^3 + 3x^2 - 7x + 5}{x + 3} = 2x^2 - 3x + 2 + \frac{-1}{x + 3}
\]
So, the coefficients are:
- \( a = 2 \)
- \( b = -3 \)
- \( c = 2 \)
- \( d = -1 \)

### Final Answer:
The values of \( a \), \( b \), \( c \), and \( d \) are \( 2 \), \( -3 \), \( 2 \), and \( -1 \) respectively.

Would you like a detailed explanation of the long division process or do you have any questions?

### Related Questions:
1. What is the general process for polynomial long division?
2. How can you verify the result of a polynomial division?
3. What are some common mistakes to avoid in polynomial long division?
4. How can synthetic division be used as an alternative to long division?
5. Why is the remainder important in polynomial division, and how does it relate to the divisor?

### Tip:
Always double-check each step in polynomial division by multiplying the divisor with the current quotient term to ensure accurate subtraction.

Learn how to solve a polynomial division problem using long division step-by-step with a detailed explanation. This problem involves dividing a polynomial by a linear divisor and finding coefficients. Suitable for advanced high school students (Grades 11-12).

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