Answer :
To find the product of the polynomials [tex]\((5x^2 - x - 3)(2x + 6)\)[/tex], we will use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial. Here are the steps:
1. Multiply [tex]\(5x^2\)[/tex] by each term in the second polynomial:
- [tex]\(5x^2 \times 2x = 10x^3\)[/tex]
- [tex]\(5x^2 \times 6 = 30x^2\)[/tex]
2. Multiply [tex]\(-x\)[/tex] by each term in the second polynomial:
- [tex]\(-x \times 2x = -2x^2\)[/tex]
- [tex]\(-x \times 6 = -6x\)[/tex]
3. Multiply [tex]\(-3\)[/tex] by each term in the second polynomial:
- [tex]\(-3 \times 2x = -6x\)[/tex]
- [tex]\(-3 \times 6 = -18\)[/tex]
4. Now, combine all these results:
[tex]\[
10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18
\][/tex]
5. Combine like terms:
- Combine [tex]\(30x^2\)[/tex] and [tex]\(-2x^2\)[/tex]: [tex]\(30x^2 - 2x^2 = 28x^2\)[/tex]
- Combine [tex]\(-6x\)[/tex] and [tex]\(-6x\)[/tex]: [tex]\(-6x - 6x = -12x\)[/tex]
6. The final result is:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
So, the correct choice is D: [tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex].
1. Multiply [tex]\(5x^2\)[/tex] by each term in the second polynomial:
- [tex]\(5x^2 \times 2x = 10x^3\)[/tex]
- [tex]\(5x^2 \times 6 = 30x^2\)[/tex]
2. Multiply [tex]\(-x\)[/tex] by each term in the second polynomial:
- [tex]\(-x \times 2x = -2x^2\)[/tex]
- [tex]\(-x \times 6 = -6x\)[/tex]
3. Multiply [tex]\(-3\)[/tex] by each term in the second polynomial:
- [tex]\(-3 \times 2x = -6x\)[/tex]
- [tex]\(-3 \times 6 = -18\)[/tex]
4. Now, combine all these results:
[tex]\[
10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18
\][/tex]
5. Combine like terms:
- Combine [tex]\(30x^2\)[/tex] and [tex]\(-2x^2\)[/tex]: [tex]\(30x^2 - 2x^2 = 28x^2\)[/tex]
- Combine [tex]\(-6x\)[/tex] and [tex]\(-6x\)[/tex]: [tex]\(-6x - 6x = -12x\)[/tex]
6. The final result is:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
So, the correct choice is D: [tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex].
