Answer :
To find the product of the polynomials [tex]\((5x^2 - x - 3)(2x + 6)\)[/tex], you can use the distributive property to multiply each term from the first polynomial by each term in the second polynomial. Here's how it works step-by-step:
1. Distribute [tex]\(5x^2\)[/tex]:
- Multiply [tex]\(5x^2\)[/tex] by [tex]\(2x\)[/tex]: [tex]\(5x^2 \times 2x = 10x^3\)[/tex]
- Multiply [tex]\(5x^2\)[/tex] by [tex]\(6\)[/tex]: [tex]\(5x^2 \times 6 = 30x^2\)[/tex]
2. Distribute [tex]\(-x\)[/tex]:
- Multiply [tex]\(-x\)[/tex] by [tex]\(2x\)[/tex]: [tex]\(-x \times 2x = -2x^2\)[/tex]
- Multiply [tex]\(-x\)[/tex] by [tex]\(6\)[/tex]: [tex]\(-x \times 6 = -6x\)[/tex]
3. Distribute [tex]\(-3\)[/tex]:
- Multiply [tex]\(-3\)[/tex] by [tex]\(2x\)[/tex]: [tex]\(-3 \times 2x = -6x\)[/tex]
- Multiply [tex]\(-3\)[/tex] by [tex]\(6\)[/tex]: [tex]\(-3 \times 6 = -18\)[/tex]
4. Combine all the products:
- After distributing, collect the terms:
[tex]\(10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18\)[/tex]
5. 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]
Putting it all together gives us the final product:
[tex]\[10x^3 + 28x^2 - 12x - 18\][/tex]
Therefore, the correct answer is:
C. [tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex]
1. Distribute [tex]\(5x^2\)[/tex]:
- Multiply [tex]\(5x^2\)[/tex] by [tex]\(2x\)[/tex]: [tex]\(5x^2 \times 2x = 10x^3\)[/tex]
- Multiply [tex]\(5x^2\)[/tex] by [tex]\(6\)[/tex]: [tex]\(5x^2 \times 6 = 30x^2\)[/tex]
2. Distribute [tex]\(-x\)[/tex]:
- Multiply [tex]\(-x\)[/tex] by [tex]\(2x\)[/tex]: [tex]\(-x \times 2x = -2x^2\)[/tex]
- Multiply [tex]\(-x\)[/tex] by [tex]\(6\)[/tex]: [tex]\(-x \times 6 = -6x\)[/tex]
3. Distribute [tex]\(-3\)[/tex]:
- Multiply [tex]\(-3\)[/tex] by [tex]\(2x\)[/tex]: [tex]\(-3 \times 2x = -6x\)[/tex]
- Multiply [tex]\(-3\)[/tex] by [tex]\(6\)[/tex]: [tex]\(-3 \times 6 = -18\)[/tex]
4. Combine all the products:
- After distributing, collect the terms:
[tex]\(10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18\)[/tex]
5. 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]
Putting it all together gives us the final product:
[tex]\[10x^3 + 28x^2 - 12x - 18\][/tex]
Therefore, the correct answer is:
C. [tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex]
