Answer :
To find the product of the polynomials [tex]\((5x^2 - x - 3)(2x + 6)\)[/tex], we'll use the distributive property (also known as the FOIL method when dealing with binomials). Here's the step-by-step process:
1. Distribute each term in the first polynomial to each term in the second polynomial:
[tex]\[
(5x^2 - x - 3)(2x + 6) = 5x^2 \cdot 2x + 5x^2 \cdot 6 - x \cdot 2x - x \cdot 6 - 3 \cdot 2x - 3 \cdot 6
\][/tex]
2. Multiply each term:
[tex]\[
= (5x^2 \cdot 2x) + (5x^2 \cdot 6) + (-x \cdot 2x) + (-x \cdot 6) + (-3 \cdot 2x) + (-3 \cdot 6)
\][/tex]
[tex]\[
= 10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18
\][/tex]
3. Combine like terms:
[tex]\[
= 10x^3 + (30x^2 - 2x^2) - (6x + 6x) - 18
\][/tex]
[tex]\[
= 10x^3 + 28x^2 - 12x - 18
\][/tex]
Thus, the product of the polynomials [tex]\( (5x^2 - x - 3)(2x + 6) \)[/tex] is [tex]\(\boldsymbol{10x^3 + 28x^2 - 12x - 18}\)[/tex].
So, the correct answer is:
A. [tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex]
1. Distribute each term in the first polynomial to each term in the second polynomial:
[tex]\[
(5x^2 - x - 3)(2x + 6) = 5x^2 \cdot 2x + 5x^2 \cdot 6 - x \cdot 2x - x \cdot 6 - 3 \cdot 2x - 3 \cdot 6
\][/tex]
2. Multiply each term:
[tex]\[
= (5x^2 \cdot 2x) + (5x^2 \cdot 6) + (-x \cdot 2x) + (-x \cdot 6) + (-3 \cdot 2x) + (-3 \cdot 6)
\][/tex]
[tex]\[
= 10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18
\][/tex]
3. Combine like terms:
[tex]\[
= 10x^3 + (30x^2 - 2x^2) - (6x + 6x) - 18
\][/tex]
[tex]\[
= 10x^3 + 28x^2 - 12x - 18
\][/tex]
Thus, the product of the polynomials [tex]\( (5x^2 - x - 3)(2x + 6) \)[/tex] is [tex]\(\boldsymbol{10x^3 + 28x^2 - 12x - 18}\)[/tex].
So, the correct answer is:
A. [tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex]
