Please simplify using long division (showing all work) : (8b^3 -6) / (2b-1).

Thank you.

.....4b^2 + 2b + 1

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2b-1 [8b^3 + 0b^2 + 0b - 6
......8b^3 - 4b^2
.............4b^2 + 0b
.............4b^2 - 2b
....................2b - 6
....................2b - 1
.........................5 remainder

To simplify the expression (8b^3 - 6) / (2b - 1) using long division, follow these steps:

Step 1: Write the expression in long division format:
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(2b - 1) | 8b^3 + 0b^2 + 0b - 6

Step 2: Divide the first term of the numerator by the first term of the denominator to get the quotient:
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(2b - 1) | 4b^2

Step 3: Multiply the quotient obtained in step 2 by the entire denominator:
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(2b - 1) | 4b^2 + 2b

Step 4: Subtract the result from step 3 from the original numerator:
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(2b - 1) | 4b^2 + 2b
- (4b^2 - 2b)

= 2b + 0b = 2b

Step 5: Bring down the next term from the numerator (-6 in this case):
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(2b - 1) | 4b^2 + 2b - 6

Step 6: Divide the new term by the first term of the denominator and write the quotient:
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(2b - 1) | 4b^2 + 2b - 6
- (4b^2 - 2b)

= 2

Step 7: Multiply the quotient obtained in step 6 by the entire denominator:
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(2b - 1) | 4b^2 + 2b - 6
- (4b^2 - 2b)
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+ 2b - 1

Step 8: Subtract the result from step 7 from the remaining terms:
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(2b - 1) | 4b^2 + 2b - 6
- (4b^2 - 2b)
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+ 2b - 1
- (2b - 1)

= -2

The final simplified expression is: 4b^2 + 2 - 2 / (2b - 1), or more commonly written as 4b^2 - 2 / (2b - 1).