What is the factored form of an expression?

1. Identify the expression: The expression to be factored is $2x^3 + 4x^2 - x$.
2. Find the greatest common factor (GCF): Examine each term in the expression to find the largest factor that divides into all of them.
- The coefficients are 2, 4, and -1. The GCF of these coefficients is 1.
- The variables are $x^3$, $x^2$, and $x$. The lowest power of $x$ present in all terms is $x^1$ or $x$.
- Therefore, the GCF of the terms is $1x$, or simply $x$.
3. Factor out the GCF: Divide each term of the original expression by the GCF ($x$).
- For the first term: $\frac{2x^3}{x} = 2x^{3-1} = 2x^2$.
- For the second term: $\frac{4x^2}{x} = 4x^{2-1} = 4x$.
- For the third term: $\frac{-x}{x} = -1$.
4. Write the factored form: The factored form of the expression is the GCF multiplied by the result of the division.
- Factored form: $x(2x^2 + 4x - 1)$.
5. Verify the answer (optional): Distribute the GCF back into the parentheses to check if it matches the original expression.
- $x \times (2x^2 + 4x - 1) = x(2x^2) + x(4x) + x(-1) = 2x^3 + 4x^2 - x$.
- The result matches the original expression, so the factorization is correct.