To find the difference between the polynomials [tex]\(\left(5 x^3 + 4 x^2\right)\)[/tex] and [tex]\(\left(6 x^2 - 2 x - 9\right)\)[/tex], we need to subtract the second polynomial from the first.
Here's a step-by-step solution:
1. Write down the polynomials:
[tex]\[
p1(x) = 5x^3 + 4x^2 + 0x + 0
\][/tex]
[tex]\[
p2(x) = 0x^3 + 6x^2 - 2x - 9
\][/tex]
2. Set up the expression for subtraction:
[tex]\[
(5x^3 + 4x^2 + 0x + 0) - (0x^3 + 6x^2 - 2x - 9)
\][/tex]
3. Distribute the subtraction across the second polynomial:
[tex]\[
5x^3 + 4x^2 + 0x + 0 - 0x^3 - 6x^2 + 2x + 9
\][/tex]
4. Combine like terms:
Let's combine the coefficients of each power of [tex]\(x\)[/tex]:
- For [tex]\(x^3\)[/tex]: [tex]\(5x^3 - 0x^3 = 5x^3\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(4x^2 - 6x^2 = -2x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(0x + 2x = 2x\)[/tex]
- For the constant term: [tex]\(0 + 9 = 9\)[/tex]
5. Write the resulting polynomial:
[tex]\[
5x^3 - 2x^2 + 2x + 9
\][/tex]
Thus, the difference of the polynomials is:
[tex]\[
5x^3 - 2x^2 + 2x + 9
\][/tex]
So, the correct choice is:
[tex]\[
5 x^3 - 2 x^2 + 2 x + 9
\][/tex]
