Write the quotient and remainder when we divide : `(2x^(3) + x^(2) - 5x - 2)` by (2x + 3)

Answer

To divide the polynomial \(2x^3 + x^2 - 5x - 2\) by \(2x + 3\), we will use polynomial long division. Here are the steps:

### Step 1: Set up the division
We will write \(2x^3 + x^2 - 5x - 2\) under the long division symbol and \(2x + 3\) outside.

### Step 2: Divide the leading terms
Divide the leading term of the dividend \(2x^3\) by the leading term of the divisor \(2x\):
\[ \frac{2x^3}{2x} = x^2 \]
This \(x^2\) is the first term of our quotient.

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

### Step 4: Repeat the process
Now, we will repeat the process with the new polynomial \(-2x^2 - 5x - 2\).

1. Divide the leading term \(-2x^2\) by \(2x\):
\[ \frac{-2x^2}{2x} = -x \]
This \(-x\) is the next term of our quotient.

2. Multiply \(-x\) by \(2x + 3\):
\[ -x(2x + 3) = -2x^2 - 3x \]
Now subtract this from \(-2x^2 - 5x - 2\):
\[ (-2x^2 - 5x - 2) - (-2x^2 - 3x) = (-5x + 3x) - 2 = -2x - 2 \]

### Step 5: Continue the process
Now, we will repeat the process with \(-2x - 2\).

1. Divide the leading term \(-2x\) by \(2x\):
\[ \frac{-2x}{2x} = -1 \]
This \(-1\) is the next term of our quotient.

2. Multiply \(-1\) by \(2x + 3\):
\[ -1(2x + 3) = -2x - 3 \]
Now subtract this from \(-2x - 2\):
\[ (-2x - 2) - (-2x - 3) = (-2 + 3) = 1 \]

### Step 6: Conclusion
We have reached a point where the degree of the remainder \(1\) is less than the degree of the divisor \(2x + 3\). Thus, we stop here.

The final results are:
- **Quotient**: \(x^2 - x - 1\)
- **Remainder**: \(1\)

### Summary
When dividing \(2x^3 + x^2 - 5x - 2\) by \(2x + 3\), the quotient is \(x^2 - x - 1\) and the remainder is \(1\).

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