To find the sum of two polynomials, you combine like terms. Here, we aim to simplify the expression:
[tex]\[ \left(9 - 3x^2\right) + \left(-8x^2 + 4x + 5\right) \][/tex]
We will break it down step-by-step:
1. Identify like terms:
- The constant terms are [tex]\(9\)[/tex] and [tex]\(5\)[/tex].
- The [tex]\(x^2\)[/tex] terms are [tex]\(-3x^2\)[/tex] and [tex]\(-8x^2\)[/tex].
- The linear term is [tex]\(4x\)[/tex].
2. Combine the like terms:
- For the [tex]\(x^2\)[/tex] terms: [tex]\( -3x^2 + (-8x^2) \)[/tex] simplifies to [tex]\(-11x^2\)[/tex].
- The linear term remains as [tex]\( 4x \)[/tex].
- For the constants: [tex]\( 9 + 5 \)[/tex] simplifies to [tex]\(14\)[/tex].
Therefore, combining all the like terms, we have:
[tex]\[ -11x^2 + 4x + 14 \][/tex]
Now, examine the given choices:
1. [tex]\(\left[\left(-3 x^2\right)+\left(-8 x^2\right)\right]+4 x+[9+(-5)]\)[/tex]
2. [tex]\(\left[3 x^2+8 x^2\right]+4 x+[9+(-5)]\)[/tex]
3. [tex]\(\left[3 x^2+\left(-8 x^2\right)\right]+4 x+[9+5]\)[/tex]
4. [tex]\(\left[\left(-3 x^2\right)+\left(-8 x^2\right)\right]+4 x+[9+5]\)[/tex]
Let's evaluate each option:
- Choice 1: [tex]\(\left[\left(-3 x^2\right)+\left(-8 x^2\right)\right]+4 x+[9+(-5)]\)[/tex]
- This combines the [tex]\(x^2\)[/tex] terms correctly but makes a mistake in the constant terms, [tex]\(9 + (-5)\)[/tex] instead of [tex]\(9 + 5\)[/tex].
- Choice 2: [tex]\(\left[3 x^2+8 x^2\right]+4 x+[9+(-5)]\)[/tex]
- This incorrectly represents the [tex]\(x^2\)[/tex] terms as positive and also sums the constant terms incorrectly.
- Choice 3: [tex]\(\left[3 x^2+\left(-8 x^2\right)\right]+4 x+[9+5]\)[/tex]
- This makes a mistake in the signs of the [tex]\(x^2\)[/tex] terms but correctly sums the constants.
- Choice 4: [tex]\(\left[\left(-3 x^2\right)+\left(-8 x^2\right)\right]+4 x+[9+5]\)[/tex]
- This combines the [tex]\(x^2\)[/tex] terms correctly, includes the [tex]\(4x\)[/tex] term, and sums the constants correctly.
Thus, the correct expression used to find the sum of the polynomials is:
[tex]\[ \left[\left(-3 x^2\right)+\left(-8 x^2\right)\right]+4 x+[9+5] \][/tex]
So, the correct choice is:
[tex]\[ \boxed{\left[\left(-3 x^2\right)+\left(-8 x^2\right)\right]+4 x+[9+5]} \][/tex]
