The given expression is \( 2x^2 - x - 1 \).

If you're looking to factor it, let's solve it step by step:

1. Multiply the coefficient of \( x^2 \) (which is 2) by the constant term (which is -1):
   \[
   2 \times -1 = -2
   \]

2. Find two numbers that multiply to give -2 and add up to the coefficient of \( x \) (which is -1). These numbers are -2 and 1 because:
   \[
   -2 \times 1 = -2 \quad \text{and} \quad -2 + 1 = -1
   \]

3. Rewrite the middle term \( -x \) as \( -2x + x \):
   \[
   2x^2 - 2x + x - 1
   \]

4. Factor by grouping:
   \[
   2x(x - 1) + 1(x - 1)
   \]

5. Factor out the common term \( (x - 1) \):
   \[
   (2x + 1)(x - 1)
   \]

So, the factored form of \( 2x^2 - x - 1 \) is:
\[
(2x + 1)(x - 1)
\]

Would you like a more detailed explanation, or do you have any other questions?

Here are five related questions:
1. How can you use the quadratic formula to solve \( 2x^2 - x - 1 = 0 \)?
2. What is the difference between factoring and completing the square?
3. How can you check if a quadratic expression is factorable?
4. What are the roots of \( (2x + 1)(x - 1) = 0 \)?
5. How do you graph the equation \( 2x^2 - x - 1 \)?

**Tip:** Always double-check your factored terms by expanding them to verify correctness.

Learn how to factor the quadratic equation 2x^2 - x - 1 with a detailed step-by-step explanation. This guide explores key algebraic concepts, including factoring by grouping and the zero product property, suitable for students in Grades 9-11.

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