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Answer :
To find the product of the polynomials [tex]\((5x^2 - x - 3)\)[/tex] and [tex]\((2x + 6)\)[/tex], we'll multiply each term in the first polynomial by each term in the second polynomial. Let's go through the steps:
1. Distribute [tex]\(5x^2\)[/tex]:
- Multiply [tex]\(5x^2\)[/tex] by each term in [tex]\((2x + 6)\)[/tex]:
- [tex]\(5x^2 \times 2x = 10x^3\)[/tex]
- [tex]\(5x^2 \times 6 = 30x^2\)[/tex]
2. Distribute [tex]\(-x\)[/tex]:
- Multiply [tex]\(-x\)[/tex] by each term in [tex]\((2x + 6)\)[/tex]:
- [tex]\(-x \times 2x = -2x^2\)[/tex]
- [tex]\(-x \times 6 = -6x\)[/tex]
3. Distribute [tex]\(-3\)[/tex]:
- Multiply [tex]\(-3\)[/tex] by each term in [tex]\((2x + 6)\)[/tex]:
- [tex]\(-3 \times 2x = -6x\)[/tex]
- [tex]\(-3 \times 6 = -18\)[/tex]
4. Combine all the results:
- Combine all these terms together:
[tex]\[
10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18
\][/tex]
5. Simplify by combining like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(30x^2 - 2x^2 = 28x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-6x - 6x = -12x\)[/tex]
Putting it all together, the product of the polynomials is:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
Thus, the correct answer is option A: [tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex].
1. Distribute [tex]\(5x^2\)[/tex]:
- Multiply [tex]\(5x^2\)[/tex] by each term in [tex]\((2x + 6)\)[/tex]:
- [tex]\(5x^2 \times 2x = 10x^3\)[/tex]
- [tex]\(5x^2 \times 6 = 30x^2\)[/tex]
2. Distribute [tex]\(-x\)[/tex]:
- Multiply [tex]\(-x\)[/tex] by each term in [tex]\((2x + 6)\)[/tex]:
- [tex]\(-x \times 2x = -2x^2\)[/tex]
- [tex]\(-x \times 6 = -6x\)[/tex]
3. Distribute [tex]\(-3\)[/tex]:
- Multiply [tex]\(-3\)[/tex] by each term in [tex]\((2x + 6)\)[/tex]:
- [tex]\(-3 \times 2x = -6x\)[/tex]
- [tex]\(-3 \times 6 = -18\)[/tex]
4. Combine all the results:
- Combine all these terms together:
[tex]\[
10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18
\][/tex]
5. Simplify by combining like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(30x^2 - 2x^2 = 28x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-6x - 6x = -12x\)[/tex]
Putting it all together, the product of the polynomials is:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
Thus, the correct answer is option A: [tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex].
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