Let's break down the expression and simplify it step by step to find which option it is equivalent to.
The given expression is:
[tex]\[ -2\left(x^2 - 2x + 1\right) + \left(3x^2 + 3x - 5\right) \][/tex]
1. Distribute the [tex]\(-2\)[/tex] through the first term:
[tex]\[ -2(x^2) -2(-2x) -2(1) \][/tex]
Which simplifies to:
[tex]\[ -2x^2 + 4x - 2 \][/tex]
2. Now we have:
[tex]\[ -2x^2 + 4x - 2 + \left(3x^2 + 3x - 5\right) \][/tex]
3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[ -2x^2 + 3x^2 = x^2 \][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[ 4x + 3x = 7x \][/tex]
- Combine the constant terms:
[tex]\[ -2 - 5 = -7 \][/tex]
4. So the simplified expression is:
[tex]\[ x^2 + 7x - 7 \][/tex]
Among the given options, this matches with:
4) [tex]\( x^2 + 7x - 7 \)[/tex]
Therefore, the expression [tex]\( -2\left(x^2-2 x+1\right) + \left(3 x^2+3 x-5\right) \)[/tex] is equivalent to [tex]\( x^2 + 7x - 7 \)[/tex]. Thus, the correct answer is:
4) [tex]\( x^2 + 7 x- 7 \)[/tex]
