Find the quotient and remainder using long division for: 
\frac{2x^3-6x^2+7x-32}{2x^2+5}

To divide \( \frac{2x^3 - 6x^2 + 7x - 32}{2x^2 + 5} \) using long division, follow these steps:

### Step 1: Divide the first terms
Divide the leading term of the dividend (\(2x^3\)) by the leading term of the divisor (\(2x^2\)):

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

### Step 2: Multiply the divisor by this quotient
Now, multiply \(x\) by the entire divisor \(2x^2 + 5\):

\[
x \cdot (2x^2 + 5) = 2x^3 + 5x
\]

### Step 3: Subtract this result from the dividend
Subtract \((2x^3 + 5x)\) from \((2x^3 - 6x^2 + 7x - 32)\):

\[
(2x^3 - 6x^2 + 7x - 32) - (2x^3 + 5x) = -6x^2 + 2x - 32
\]

### Step 4: Divide the new leading term
Next, divide the leading term of the new expression (\(-6x^2\)) by the leading term of the divisor (\(2x^2\)):

\[
\frac{-6x^2}{2x^2} = -3
\]

### Step 5: Multiply the divisor by this quotient
Multiply \(-3\) by the entire divisor \(2x^2 + 5\):

\[
-3 \cdot (2x^2 + 5) = -6x^2 - 15
\]

### Step 6: Subtract this result from the new expression
Subtract \((-6x^2 - 15)\) from \((-6x^2 + 2x - 32)\):

\[
(-6x^2 + 2x - 32) - (-6x^2 - 15) = 2x - 17
\]

### Step 7: Write the quotient and remainder
At this point, the quotient is \(x - 3\), and the remainder is \(2x - 17\).

Thus, the final answer is:

\[
\frac{2x^3 - 6x^2 + 7x - 32}{2x^2 + 5} = x - 3 + \frac{2x - 17}{2x^2 + 5}
\]

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### Related Questions:
1. How can I apply long division to a higher-degree polynomial?
2. What are the uses of polynomial division in calculus or algebra?
3. Can you explain how to handle remainders in polynomial division?
4. How is synthetic division different from long division?
5. What happens if the remainder is zero?

**Tip:** Always check if terms are arranged in descending powers of \(x\) before starting polynomial division!

Find the quotient and remainder using long division for: \frac{2x^3 - 6x^2 + 7x - 32}{2x^2 + 5}

Learn how to divide the polynomial 2x^3 - 6x^2 + 7x - 32 by 2x^2 + 5 using long division. This step-by-step guide walks you through dividing the leading terms, multiplying, subtracting, and finding the quotient and remainder. Ideal for high school students (Grades 9-12) studying algebra and polynomial division.

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