To determine the polynomial that represents the difference between the two given polynomials [tex]\(5x^2 + 9x + 3\)[/tex] and [tex]\(6x^2 - 3x\)[/tex], follow these steps:
1. Write down the original polynomials:
- Polynomial 1: [tex]\(5x^2 + 9x + 3\)[/tex]
- Polynomial 2: [tex]\(6x^2 - 3x\)[/tex]
2. Set up the subtraction:
[tex]\[
(5x^2 + 9x + 3) - (6x^2 - 3x)
\][/tex]
3. Distribute the negative sign through the second polynomial:
[tex]\[
5x^2 + 9x + 3 - 6x^2 + 3x
\][/tex]
4. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(5x^2 - 6x^2 = -x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(9x + 3x = 12x\)[/tex]
- The constant terms remain as they are since there is no constant term in the second polynomial to combine: [tex]\(3\)[/tex]
5. Write the resulting polynomial:
[tex]\[
-x^2 + 12x + 3
\][/tex]
Thus, the polynomial that represents the difference is:
[tex]\[ -x^2 + 12x + 3 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{-x^2 + 12x + 3} \][/tex]
