Certainly! Let's analyze each polynomial operation step-by-step.
### 1. [tex]\(\left(5x^2 + 2x + 1\right) + \left(6x^2 - 3x + 9\right)\)[/tex]
Combine like terms:
- [tex]\(5x^2 + 6x^2 = 11x^2\)[/tex]
- [tex]\(2x - 3x = -x\)[/tex]
- [tex]\(1 + 9 = 10\)[/tex]
The resulting expression is:
[tex]\[11x^2 - x + 10\][/tex]
This corresponds to expression B.
### 2. [tex]\(\left(-3x^2 + 6x - 12\right) + (5x + 9)\)[/tex]
Combine like terms:
- [tex]\(-3x^2\)[/tex] (no other [tex]\(x^2\)[/tex] term to combine with)
- [tex]\(6x + 5x = 11x\)[/tex]
- [tex]\(-12 + 9 = -3\)[/tex]
The resulting expression is:
[tex]\[-3x^2 + 11x - 3\][/tex]
This corresponds to expression A.
### 3. [tex]\((8x + 16) - \left(3x^2 - 3x - 15\right)\)[/tex]
Distribute the negative sign and combine like terms:
- [tex]\(-3x^2\)[/tex] (no other [tex]\(x^2\)[/tex] term to combine with)
- [tex]\(8x + 3x = 11x\)[/tex]
- [tex]\(16 + 15 = 31\)[/tex]
The resulting expression is:
[tex]\[-3x^2 + 11x + 31\][/tex]
This corresponds to expression D.
### 4. [tex]\(\left(5x^2 + 23x - 7\right) - \left(x^2 - 4x + 21\right)\)[/tex]
Distribute the negative sign and combine like terms:
- [tex]\(5x^2 - x^2 = 4x^2\)[/tex]
- [tex]\(23x + 4x = 27x\)[/tex]
- [tex]\(-7 - 21 = -28\)[/tex]
The resulting expression is:
[tex]\[4x^2 + 27x - 28\][/tex]
This corresponds to expression C.
Summarizing all parts together:
- [tex]\(\left(5x^2 + 2x + 1\right) + \left(6x^2 - 3x + 9\right)\)[/tex] is equivalent to expression B
- [tex]\(\left(-3x^2 + 6x - 12\right) + (5x + 9)\)[/tex] is equivalent to expression A
- [tex]\((8x + 16) - \left(3x^2 - 3x - 15\right)\)[/tex] is equivalent to expression D
- [tex]\(\left(5x^2 + 23x - 7\right) - \left(x^2 - 4x + 21\right)\)[/tex] is equivalent to expression C
