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We are tasked with performing polynomial long division to divide \( 6x^3 - 29x^2 + 16x + 21 \) by \( 2x - 3 \). Let's go step by step through the long division process:

### Step 1: Set up the division
We are dividing \( 6x^3 - 29x^2 + 16x + 21 \) by \( 2x - 3 \).

- First, divide the leading term of the dividend (\( 6x^3 \)) by the leading term of the divisor (\( 2x \)):
  \[
  \frac{6x^3}{2x} = 3x^2
  \]
  This is the first term of the quotient.

### Step 2: Multiply and subtract
Now multiply \( 3x^2 \) by \( 2x - 3 \):
\[
3x^2 \times (2x - 3) = 6x^3 - 9x^2
\]
Subtract this from the original polynomial:
\[
(6x^3 - 29x^2 + 16x + 21) - (6x^3 - 9x^2) = -20x^2 + 16x + 21
\]

### Step 3: Repeat the process
Now, divide the new leading term (\( -20x^2 \)) by the leading term of the divisor (\( 2x \)):
\[
\frac{-20x^2}{2x} = -10x
\]
This is the second term of the quotient.

Multiply \( -10x \) by \( 2x - 3 \):
\[
-10x \times (2x - 3) = -20x^2 + 30x
\]
Subtract:
\[
(-20x^2 + 16x + 21) - (-20x^2 + 30x) = -14x + 21
\]

### Step 4: Continue dividing
Now divide \( -14x \) by \( 2x \):
\[
\frac{-14x}{2x} = -7
\]
This is the third term of the quotient.

Multiply \( -7 \) by \( 2x - 3 \):
\[
-7 \times (2x - 3) = -14x + 21
\]
Subtract:
\[
(-14x + 21) - (-14x + 21) = 0
\]

### Final Result
Thus, the quotient is:
\[
3x^2 - 10x - 7
\]
And the remainder is 0.

Would you like further details or have any questions?

### 5 Related Questions:
1. How would the result change if we divided by a different binomial, like \( x + 2 \)?
2. How does synthetic division compare to long division in this context?
3. Can you use the remainder theorem to verify that the remainder is 0?
4. How do the degrees of the polynomials affect the quotient and remainder in long division?
5. Can this process be applied to dividing by higher-degree polynomials?

### Tip:
Always ensure to align the terms by their degrees when performing long division to avoid errors in subtraction.

Use the long division method to find the result when 6x^3 - 29x^2 + 16x + 21 is divided by 2x - 3.

Learn how to solve the polynomial division of 6x^3 - 29x^2 + 16x + 21 by 2x - 3 using the long division method. This problem walks through key steps in polynomial division, suitable for students in Grades 10-12.

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