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Answer :
To find the product of the given polynomials [tex]\((5x^2 - x - 3)(2x + 6)\)[/tex], we will multiply each term in the first polynomial by each term in the second polynomial. Here's how you can do the multiplication step-by-step:
1. Multiply the first term of the first polynomial by each term of the second polynomial:
[tex]\[
5x^2 \times 2x = 10x^3
\][/tex]
[tex]\[
5x^2 \times 6 = 30x^2
\][/tex]
2. Multiply the second term of the first polynomial by each term of the second polynomial:
[tex]\[
(-x) \times 2x = -2x^2
\][/tex]
[tex]\[
(-x) \times 6 = -6x
\][/tex]
3. Multiply the third term of the first polynomial by each term of the second polynomial:
[tex]\[
(-3) \times 2x = -6x
\][/tex]
[tex]\[
(-3) \times 6 = -18
\][/tex]
4. Combine all these results:
- [tex]\(10x^3\)[/tex] (from [tex]\(5x^2 \times 2x\)[/tex])
- [tex]\(30x^2\)[/tex] and [tex]\(-2x^2\)[/tex] add up to [tex]\(28x^2\)[/tex] (from [tex]\(5x^2 \times 6\)[/tex] and [tex]\((-x) \times 2x\)[/tex])
- [tex]\(-6x\)[/tex] and [tex]\(-6x\)[/tex] add up to [tex]\(-12x\)[/tex] (from [tex]\((-x) \times 6\)[/tex] and [tex]\((-3) \times 2x\)[/tex])
- [tex]\(-18\)[/tex] (from [tex]\((-3) \times 6\)[/tex])
So, the product of the polynomials is:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
The correct answer is Option B: [tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex].
1. Multiply the first term of the first polynomial by each term of the second polynomial:
[tex]\[
5x^2 \times 2x = 10x^3
\][/tex]
[tex]\[
5x^2 \times 6 = 30x^2
\][/tex]
2. Multiply the second term of the first polynomial by each term of the second polynomial:
[tex]\[
(-x) \times 2x = -2x^2
\][/tex]
[tex]\[
(-x) \times 6 = -6x
\][/tex]
3. Multiply the third term of the first polynomial by each term of the second polynomial:
[tex]\[
(-3) \times 2x = -6x
\][/tex]
[tex]\[
(-3) \times 6 = -18
\][/tex]
4. Combine all these results:
- [tex]\(10x^3\)[/tex] (from [tex]\(5x^2 \times 2x\)[/tex])
- [tex]\(30x^2\)[/tex] and [tex]\(-2x^2\)[/tex] add up to [tex]\(28x^2\)[/tex] (from [tex]\(5x^2 \times 6\)[/tex] and [tex]\((-x) \times 2x\)[/tex])
- [tex]\(-6x\)[/tex] and [tex]\(-6x\)[/tex] add up to [tex]\(-12x\)[/tex] (from [tex]\((-x) \times 6\)[/tex] and [tex]\((-3) \times 2x\)[/tex])
- [tex]\(-18\)[/tex] (from [tex]\((-3) \times 6\)[/tex])
So, the product of the polynomials is:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
The correct answer is Option B: [tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex].
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