Let’s simplify the expression step by step. First, we combine like terms:
- 2x² (the x² term)
- 2x + 2x + 3x + 2x + 6x + 4x + 2x + 3x = 22x (the x term)
- We have a constant term: -24 + 4 + 6 + 8 = -6
This gives us:
2x² + 22x – 6
Next, we can factor out the greatest common factor (GCF), which is 2:
2(x² + 11x – 3)
Now we need to factor the quadratic expression within the parentheses. To factor x² + 11x – 3, we look for two numbers that multiply to -3 and add to 11. However, the simplest approach here is to use the quadratic formula because it does not easily factor:
x = [−b ± sqrt(b² – 4ac)] / 2a
For a = 1, b = 11, and c = -3:
x = [−11 ± sqrt(11² – 4(1)(−3))] / 2(1)
x = [−11 ± sqrt(121 + 12)] / 2
x = [−11 ± sqrt(133)] / 2
Since the quadratic does not factor nicely, we can leave the factored form as:
2(x² + 11x – 3)
In conclusion, the completely factored form is:
2(x² + 11x – 3)