Sure, let's address each step-by-step and match the polynomials.
1. Add the polynomials: [tex]\( \left(-4x^2 - 3x + 6\right) + \left(6x^2 - 2x + 1\right) \)[/tex]
We need to add the corresponding coefficients:
[tex]\[
-4x^2 + 6x^2 = 2x^2
\][/tex]
[tex]\[
-3x - 2x = -5x
\][/tex]
[tex]\[
6 + 1 = 7
\][/tex]
So the result is:
[tex]\[
2x^2 - 5x + 7
\][/tex]
Match: [tex]\( 2x^2 - 5x + 7 \)[/tex]
2. Find the opposite of: [tex]\( 2x^2 + x - 7 \)[/tex]
The opposite polynomial means changing the sign of each term:
[tex]\[
-2x^2 - x + 7
\][/tex]
Match: [tex]\( -2x^2 - x + 7 \)[/tex]
3. Subtract the polynomials: [tex]\( \left(-4x^2 + 2x - 1\right) - \left(-2x^2 + 3x + 6\right) \)[/tex]
First, distribute the negative sign to the second polynomial:
[tex]\[
-(-2x^2 + 3x + 6) = 2x^2 - 3x - 6
\][/tex]
Now, add this to the first polynomial:
[tex]\[
(-4x^2 + 2x - 1) + (2x^2 - 3x - 6)
\][/tex]
Combine the like terms:
[tex]\[
-4x^2 + 2x^2 = -2x^2
\][/tex]
[tex]\[
2x - 3x = -1x
\][/tex]
[tex]\[
-1 - 6 = -7
\][/tex]
So, the result is:
[tex]\[
-2x^2 - x - 7
\][/tex]
Match: [tex]\( -2x^2 - x - 7 \)[/tex]
Grouping the results and available options:
- [tex]\( 2x^2 - 5x + 7 \)[/tex] matches [tex]\( 2 x^2 - 5 x+7 \)[/tex]
- [tex]\( -2x^2 - x + 7 \)[/tex] matches [tex]\( -2 x^2-x+7 \)[/tex]
- [tex]\( -2x^2 - x - 7 \)[/tex] matches [tex]\( -2 x^2-x-7 \)[/tex]
