To determine the correct expression for finding the sum of the polynomials \( \left(9 - 3x^2\right) + \left(-8x^2 + 4x + 5\right) \), we need to sum the corresponding coefficients of the terms.
Let's break down the problem step by step:
1. Identify and sum the constant terms:
- From the first polynomial \(9 - 3x^2\), the constant term is \(9\).
- From the second polynomial \(-8x^2 + 4x + 5\), the constant term is \(5\).
Adding the constant terms:
[tex]\[
9 + 5 = 14
\][/tex]
2. Identify and sum the linear terms (the coefficients of \(x\)):
- From the first polynomial \(9 - 3x^2\), there is no linear term (the coefficient of \(x\) is \(0\)).
- From the second polynomial \(-8x^2 + 4x + 5\), the linear term is \(4x\).
Adding the linear terms:
[tex]\[
0 + 4 = 4
\][/tex]
So, the linear term in the sum is \(4x\).
3. Identify and sum the quadratic terms (the coefficients of \(x^2\)):
- From the first polynomial \( 9 - 3x^2\), the quadratic term is \(-3x^2\).
- From the second polynomial \(-8x^2 + 4x + 5\), the quadratic term is \(-8x^2\).
Adding the quadratic terms:
[tex]\[
-3x^2 + (-8x^2) = -11x^2
\][/tex]
Putting it all together, the sum of the polynomials is:
[tex]\[
14 + 4x - 11x^2
\][/tex]
Now, let’s match this result with the given options:
1. \(\left(9 - 3x^2\right) + \left(-8x^2 + 4x + 5\right)\)
2. \(\left[\left(-3x^2\right) + \left(-8x^2\right)\right] + 4x + [9 + (-5)]\)
- Notice that this sums the quadratic terms incorrectly as \(-3x^2 + (-8x^2)\), which is correct.
- However, it combines the constants incorrectly: \( 9 + (-5) \).
3. \(\left[3x^2 + 8x^2\right] + 4x + [9 + (-5)]\)
- This option sums the quadratic terms as \(3x^2 + 8x^2\), which is incorrect.
4. \(\left[3x^2 + \left(-8x^2\right)\right] + 4x + [9 + 5]\)
- This option sums the quadratic terms as \(3x^2 + (-8x^2)\), which is incorrect, the sum should be \(-3x^2 + (-8x^2)\).
5. \(\left[\left(-3x^2\right) + \left(-8x^2\right)\right] + 4x + [9 + 5]\)
- This option sums the quadratic terms correctly: \(-3x^2 + (-8x^2) = -11x^2\).
- It sums the constants correctly: \(9 + 5 = 14\).
Thus, the correct answer is:
[tex]\[
\boxed{4}
\][/tex]
