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Answer :
Let's find the product of the polynomials [tex]\((5x^2 - x - 3)\)[/tex] and [tex]\((2x + 6)\)[/tex] step by step.
1. Distribute Each Term:
- First, distribute [tex]\(5x^2\)[/tex] from [tex]\((5x^2 - x - 3)\)[/tex] to each term in [tex]\((2x + 6)\)[/tex]:
[tex]\[
5x^2 \cdot 2x = 10x^3
\][/tex]
[tex]\[
5x^2 \cdot 6 = 30x^2
\][/tex]
- Next, distribute [tex]\(-x\)[/tex] to each term in [tex]\((2x + 6)\)[/tex]:
[tex]\[
-x \cdot 2x = -2x^2
\][/tex]
[tex]\[
-x \cdot 6 = -6x
\][/tex]
- Finally, distribute [tex]\(-3\)[/tex] to each term in [tex]\((2x + 6)\)[/tex]:
[tex]\[
-3 \cdot 2x = -6x
\][/tex]
[tex]\[
-3 \cdot 6 = -18
\][/tex]
2. Combine Like Terms:
- Combine all of the resulting terms to form the expanded polynomial:
[tex]\[
10x^3 + (30x^2 - 2x^2) + (-6x - 6x) - 18
\][/tex]
- Simplify by combining like terms:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
The product of the polynomials is:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
So, the correct answer is option C:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
1. Distribute Each Term:
- First, distribute [tex]\(5x^2\)[/tex] from [tex]\((5x^2 - x - 3)\)[/tex] to each term in [tex]\((2x + 6)\)[/tex]:
[tex]\[
5x^2 \cdot 2x = 10x^3
\][/tex]
[tex]\[
5x^2 \cdot 6 = 30x^2
\][/tex]
- Next, distribute [tex]\(-x\)[/tex] to each term in [tex]\((2x + 6)\)[/tex]:
[tex]\[
-x \cdot 2x = -2x^2
\][/tex]
[tex]\[
-x \cdot 6 = -6x
\][/tex]
- Finally, distribute [tex]\(-3\)[/tex] to each term in [tex]\((2x + 6)\)[/tex]:
[tex]\[
-3 \cdot 2x = -6x
\][/tex]
[tex]\[
-3 \cdot 6 = -18
\][/tex]
2. Combine Like Terms:
- Combine all of the resulting terms to form the expanded polynomial:
[tex]\[
10x^3 + (30x^2 - 2x^2) + (-6x - 6x) - 18
\][/tex]
- Simplify by combining like terms:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
The product of the polynomials is:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
So, the correct answer is option C:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
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