Answer :
To find the product of the polynomials [tex]\(\left(5x^2 - x - 3\right)\)[/tex] and [tex]\(\left(2x + 6\right)\)[/tex], we will use the distributive property, also known as the FOIL method for binomials. Here's how you can multiply these two polynomials step-by-step:
1. Distribute each term from the first polynomial to each term of the second polynomial:
- Multiply [tex]\(5x^2\)[/tex] by each term in the second polynomial [tex]\((2x + 6)\)[/tex]:
- [tex]\(5x^2 \times 2x = 10x^3\)[/tex]
- [tex]\(5x^2 \times 6 = 30x^2\)[/tex]
- Multiply [tex]\(-x\)[/tex] by each term in the second polynomial [tex]\((2x + 6)\)[/tex]:
- [tex]\(-x \times 2x = -2x^2\)[/tex]
- [tex]\(-x \times 6 = -6x\)[/tex]
- Multiply [tex]\(-3\)[/tex] by each term in the second polynomial [tex]\((2x + 6)\)[/tex]:
- [tex]\(-3 \times 2x = -6x\)[/tex]
- [tex]\(-3 \times 6 = -18\)[/tex]
2. Combine 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]
So the expanded polynomial product is:
[tex]\[10x^3 + 28x^2 - 12x - 18\][/tex]
Thus, the correct answer to the product of the given polynomials is option D: [tex]\(10x^3 + 28x^2 -12x - 18\)[/tex].
1. Distribute each term from the first polynomial to each term of the second polynomial:
- Multiply [tex]\(5x^2\)[/tex] by each term in the second polynomial [tex]\((2x + 6)\)[/tex]:
- [tex]\(5x^2 \times 2x = 10x^3\)[/tex]
- [tex]\(5x^2 \times 6 = 30x^2\)[/tex]
- Multiply [tex]\(-x\)[/tex] by each term in the second polynomial [tex]\((2x + 6)\)[/tex]:
- [tex]\(-x \times 2x = -2x^2\)[/tex]
- [tex]\(-x \times 6 = -6x\)[/tex]
- Multiply [tex]\(-3\)[/tex] by each term in the second polynomial [tex]\((2x + 6)\)[/tex]:
- [tex]\(-3 \times 2x = -6x\)[/tex]
- [tex]\(-3 \times 6 = -18\)[/tex]
2. Combine 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]
So the expanded polynomial product is:
[tex]\[10x^3 + 28x^2 - 12x - 18\][/tex]
Thus, the correct answer to the product of the given polynomials is option D: [tex]\(10x^3 + 28x^2 -12x - 18\)[/tex].
