Answer :
Sure! Let's break down the process of finding the product of the polynomials [tex]\( (5x^2 - x - 3)(2x + 6) \)[/tex] step-by-step.
First, we apply the distributive property, which involves multiplying each term in the first polynomial by each term in the second polynomial.
1. Multiply [tex]\( 5x^2 \)[/tex] with each term in [tex]\( 2x + 6 \)[/tex]:
[tex]\[
5x^2 \cdot 2x = 10x^3
\][/tex]
[tex]\[
5x^2 \cdot 6 = 30x^2
\][/tex]
2. Multiply [tex]\( -x \)[/tex] with each term in [tex]\( 2x + 6 \)[/tex]:
[tex]\[
-x \cdot 2x = -2x^2
\][/tex]
[tex]\[
-x \cdot 6 = -6x
\][/tex]
3. Multiply [tex]\( -3 \)[/tex] with each term in [tex]\( 2x + 6 \)[/tex]:
[tex]\[
-3 \cdot 2x = -6x
\][/tex]
[tex]\[
-3 \cdot 6 = -18
\][/tex]
Now, combine all these results:
[tex]\[
10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18
\][/tex]
Next, we combine like terms:
[tex]\[
10x^3 + (30x^2 - 2x^2) + (-6x - 6x) - 18
\][/tex]
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
So, the product of the polynomials [tex]\( (5x^2 - x - 3) \)[/tex] and [tex]\( (2x + 6) \)[/tex] is:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{10x^3 + 28x^2 - 12x - 18}
\][/tex]
So, the correct option from the provided choices is:
D. [tex]\( 10x^3 + 28x^2 - 12x - 18 \)[/tex]
First, we apply the distributive property, which involves multiplying each term in the first polynomial by each term in the second polynomial.
1. Multiply [tex]\( 5x^2 \)[/tex] with each term in [tex]\( 2x + 6 \)[/tex]:
[tex]\[
5x^2 \cdot 2x = 10x^3
\][/tex]
[tex]\[
5x^2 \cdot 6 = 30x^2
\][/tex]
2. Multiply [tex]\( -x \)[/tex] with each term in [tex]\( 2x + 6 \)[/tex]:
[tex]\[
-x \cdot 2x = -2x^2
\][/tex]
[tex]\[
-x \cdot 6 = -6x
\][/tex]
3. Multiply [tex]\( -3 \)[/tex] with each term in [tex]\( 2x + 6 \)[/tex]:
[tex]\[
-3 \cdot 2x = -6x
\][/tex]
[tex]\[
-3 \cdot 6 = -18
\][/tex]
Now, combine all these results:
[tex]\[
10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18
\][/tex]
Next, we combine like terms:
[tex]\[
10x^3 + (30x^2 - 2x^2) + (-6x - 6x) - 18
\][/tex]
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
So, the product of the polynomials [tex]\( (5x^2 - x - 3) \)[/tex] and [tex]\( (2x + 6) \)[/tex] is:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{10x^3 + 28x^2 - 12x - 18}
\][/tex]
So, the correct option from the provided choices is:
D. [tex]\( 10x^3 + 28x^2 - 12x - 18 \)[/tex]
