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The problem shown in the image involves long division of polynomials. The task is to divide \(6x^3 - 8x + 5\) by \(2x - 4\) using long division. Here's how you can solve it step by step:

### Step 1: Setup the long division
We are dividing \(6x^3 - 8x + 5\) by \(2x - 4\).

1. Divide the leading term of the dividend \(6x^3\) by the leading term of the divisor \(2x\): 
   \[
   \frac{6x^3}{2x} = 3x^2
   \]
   This will be the first term of the quotient.

### Step 2: Multiply and subtract
Next, multiply the entire divisor \(2x - 4\) by \(3x^2\):
\[
(2x - 4)(3x^2) = 6x^3 - 12x^2
\]
Now, subtract this from the original polynomial:
\[
(6x^3 - 8x + 5) - (6x^3 - 12x^2) = 12x^2 - 8x + 5
\]
The \(6x^3\) cancels out, leaving \(12x^2 - 8x + 5\).

### Step 3: Divide the next term
Now divide the leading term \(12x^2\) by \(2x\):
\[
\frac{12x^2}{2x} = 6x
\]
This is the next term of the quotient.

### Step 4: Multiply and subtract again
Multiply the divisor by \(6x\):
\[
(2x - 4)(6x) = 12x^2 - 24x
\]
Now subtract this from the current polynomial:
\[
(12x^2 - 8x + 5) - (12x^2 - 24x) = 16x + 5
\]
The \(12x^2\) cancels, leaving \(16x + 5\).

### Step 5: Divide again
Now divide the leading term \(16x\) by \(2x\):
\[
\frac{16x}{2x} = 8
\]
This is the final term of the quotient.

### Step 6: Multiply and subtract one last time
Multiply the divisor by \(8\):
\[
(2x - 4)(8) = 16x - 32
\]
Subtract this from the current polynomial:
\[
(16x + 5) - (16x - 32) = 37
\]
So, the remainder is \(37\).

### Final Answer
The quotient is:
\[
3x^2 + 6x + 8
\]
with a remainder of \(37\). Therefore, the full expression is:
\[
\frac{6x^3 - 8x + 5}{2x - 4} = 3x^2 + 6x + 8 + \frac{37}{2x - 4}
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you handle remainders in polynomial division?
2. What are other methods to divide polynomials besides long division?
3. How can you check if the quotient you obtained is correct?
4. How does synthetic division differ from long division for polynomials?
5. What are some real-world applications of polynomial division?

**Tip:** When dividing polynomials, always align the terms by their degree for clearer subtraction and to avoid mistakes.

Complete the long division. Divide (6x^3 - 8x + 5) by (2x - 4). Enter the terms that belong in each box.

Learn how to divide polynomials using long division. This step-by-step guide shows how to divide (6x^3 - 8x + 5) by (2x - 4) and how to handle remainders in polynomial division. Ideal for Grades 10-12.

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