Answer :
Sure! Let's find the product of the polynomials [tex]\((5x^2 - x - 3)(2x + 6)\)[/tex] step-by-step.
To do this, we'll use the distributive property, which means each term in the first polynomial should be multiplied by each term in the second polynomial.
1. Distribute [tex]\(5x^2\)[/tex]:
[tex]\[
5x^2 \times 2x = 10x^3
\][/tex]
[tex]\[
5x^2 \times 6 = 30x^2
\][/tex]
2. Distribute [tex]\(-x\)[/tex]:
[tex]\[
-x \times 2x = -2x^2
\][/tex]
[tex]\[
-x \times 6 = -6x
\][/tex]
3. Distribute [tex]\(-3\)[/tex]:
[tex]\[
-3 \times 2x = -6x
\][/tex]
[tex]\[
-3 \times 6 = -18
\][/tex]
Now, combine all these results by adding them together:
[tex]\[
10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18
\][/tex]
4. 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 polynomial simplifies to:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
Therefore, the correct answer is:
C. [tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex]
To do this, we'll use the distributive property, which means each term in the first polynomial should be multiplied by each term in the second polynomial.
1. Distribute [tex]\(5x^2\)[/tex]:
[tex]\[
5x^2 \times 2x = 10x^3
\][/tex]
[tex]\[
5x^2 \times 6 = 30x^2
\][/tex]
2. Distribute [tex]\(-x\)[/tex]:
[tex]\[
-x \times 2x = -2x^2
\][/tex]
[tex]\[
-x \times 6 = -6x
\][/tex]
3. Distribute [tex]\(-3\)[/tex]:
[tex]\[
-3 \times 2x = -6x
\][/tex]
[tex]\[
-3 \times 6 = -18
\][/tex]
Now, combine all these results by adding them together:
[tex]\[
10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18
\][/tex]
4. 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 polynomial simplifies to:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
Therefore, the correct answer is:
C. [tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex]
