Divide.
left parenthesis 2 x squared minus 7 x minus 6 right parenthesisdivided byleft parenthesis 2 x plus 1 right parenthesis
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Part 1
left parenthesis 2 x squared minus 7 x minus 6 right parenthesisdivided byleft parenthesis 2 x plus 1 right parenthesisequals
  
x minus 4 minus StartFraction 2 Over 2 x plus 1 EndFraction
 ​(Simplify your​ answer.)

We are asked to divide \( (2x^2 - 7x - 6) \) by \( (2x + 1) \) and simplify the expression. The result is already given as:

\[
(2x^2 - 7x - 6) \div (2x + 1) = x - 4 - \frac{2}{2x + 1}
\]

Let's verify this step by step using polynomial long division.

### Step 1: Set up the division

We are dividing \( 2x^2 - 7x - 6 \) by \( 2x + 1 \).

### Step 2: Perform polynomial long division

1. Divide the leading term of the dividend \( 2x^2 \) by the leading term of the divisor \( 2x \):
   \[
   \frac{2x^2}{2x} = x
   \]
   So, the first term in the quotient is \( x \).

2. Multiply \( x \) by the divisor \( 2x + 1 \):
   \[
   x(2x + 1) = 2x^2 + x
   \]
   Subtract this from the original polynomial:
   \[
   (2x^2 - 7x - 6) - (2x^2 + x) = -8x - 6
   \]

3. Divide the new leading term \( -8x \) by \( 2x \):
   \[
   \frac{-8x}{2x} = -4
   \]
   So, the next term in the quotient is \( -4 \).

4. Multiply \( -4 \) by the divisor \( 2x + 1 \):
   \[
   -4(2x + 1) = -8x - 4
   \]
   Subtract this from the current polynomial:
   \[
   (-8x - 6) - (-8x - 4) = -2
   \]

### Step 3: Write the quotient and remainder

The quotient is \( x - 4 \), and the remainder is \( -2 \). Therefore, the division can be written as:
\[
\frac{2x^2 - 7x - 6}{2x + 1} = x - 4 - \frac{2}{2x + 1}
\]
which matches the given expression.

### Final Answer:
\[
(2x^2 - 7x - 6) \div (2x + 1) = x - 4 - \frac{2}{2x + 1}
\]

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you perform synthetic division for dividing polynomials?
2. What is the difference between polynomial long division and synthetic division?
3. Can you factor \( 2x^2 - 7x - 6 \) and verify the result by factoring?
4. What other methods can you use to simplify rational expressions?
5. How can we check the division result by multiplying the quotient and the divisor?

**Tip:** Always verify the result of a division by multiplying the quotient by the divisor and adding the remainder if necessary. This helps ensure accuracy.

Divide (2x^2 - 7x - 6) by (2x + 1) and simplify the answer.

Learn how to divide the polynomial (2x^2 - 7x - 6) by (2x + 1) using polynomial long division. This step-by-step guide simplifies the expression to x - 4 - 2/(2x + 1), covering key algebraic concepts for students in Grades 9-11.

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