factor out greatest common factor 2x(-5x^2-26x-5)

To factor out the greatest common factor (GCF) from the expression \( 2x(-5x^2 - 26x - 5) \), follow these steps:

1. **Identify the GCF**:  
   The GCF in this expression is \( 2x \), which is already factored out in front of the parentheses.

2. **Write the expression with the GCF factored out**:  
   \[
   2x(-5x^2 - 26x - 5)
   \]

3. **Factor the remaining quadratic expression**:  
   To factor the quadratic expression \( -5x^2 - 26x - 5 \), first, observe that you can factor out a \(-1\) to simplify the factoring process:
   \[
   -1(5x^2 + 26x + 5)
   \]
   Now, factor \( 5x^2 + 26x + 5 \):
   - Multiply the leading coefficient (5) by the constant term (5): \( 5 \times 5 = 25 \).
   - Find two numbers that multiply to 25 and add to 26. These numbers are 25 and 1.
   - Rewrite the middle term using these numbers:
   \[
   5x^2 + 25x + x + 5
   \]
   - Factor by grouping:
   \[
   5x(x + 5) + 1(x + 5)
   \]
   - Factor out the common factor \( x + 5 \):
   \[
   (5x + 1)(x + 5)
   \]

4. **Substitute back**:  
   Now, place this back into the original expression with the GCF:
   \[
   2x \cdot (-1) \cdot (5x + 1)(x + 5)
   \]

5. **Simplify**:  
   Combine the constants:
   \[
   -2x(5x + 1)(x + 5)
   \]

So, the fully factored expression is:
\[
\boxed{-2x(5x + 1)(x + 5)}
\]

Would you like more details or have any questions?

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Here are 5 related questions:
1. How do you determine the GCF in a polynomial?
2. What is the process of factoring by grouping?
3. How do you factor a quadratic trinomial?
4. Can the quadratic formula be used in place of factoring?
5. How do you check if a factorization is correct?

**Tip:** Always double-check by expanding the factors to ensure the original expression is correctly factored.

Learn how to factor out the greatest common factor (GCF) from the expression 2x(-5x^2 - 26x - 5) with a step-by-step solution. This problem covers factoring techniques in algebra, specifically focusing on quadratic expressions and GCF. Suitable for high school students in Grades 9-12.

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