Answer :
To solve the multiplication of the polynomials [tex]\((4x^2 + 7x)(5x^2 - 3x)\)[/tex], follow these steps:
1. Distribute Each Term:
- First, distribute the [tex]\(4x^2\)[/tex] from the first polynomial across each term in the second polynomial:
[tex]\[
4x^2 \cdot 5x^2 = 20x^4
\][/tex]
[tex]\[
4x^2 \cdot (-3x) = -12x^3
\][/tex]
- Next, distribute the [tex]\(7x\)[/tex] from the first polynomial across each term in the second polynomial:
[tex]\[
7x \cdot 5x^2 = 35x^3
\][/tex]
[tex]\[
7x \cdot (-3x) = -21x^2
\][/tex]
2. Combine Like Terms:
- Now, add all these terms together:
[tex]\[
20x^4 + (-12x^3) + 35x^3 + (-21x^2)
\][/tex]
- Combine the [tex]\(x^3\)[/tex] terms:
[tex]\[
-12x^3 + 35x^3 = 23x^3
\][/tex]
- Thus, the combined polynomial is:
[tex]\[
20x^4 + 23x^3 - 21x^2
\][/tex]
3. Select the Correct Answer:
- From the choices provided, the polynomial that matches our result is Option C: [tex]\(20x^4 + 23x^3 - 21x^2\)[/tex].
Therefore, the correct answer is:
C. [tex]\(20x^4 + 23x^3 - 21x^2\)[/tex]
1. Distribute Each Term:
- First, distribute the [tex]\(4x^2\)[/tex] from the first polynomial across each term in the second polynomial:
[tex]\[
4x^2 \cdot 5x^2 = 20x^4
\][/tex]
[tex]\[
4x^2 \cdot (-3x) = -12x^3
\][/tex]
- Next, distribute the [tex]\(7x\)[/tex] from the first polynomial across each term in the second polynomial:
[tex]\[
7x \cdot 5x^2 = 35x^3
\][/tex]
[tex]\[
7x \cdot (-3x) = -21x^2
\][/tex]
2. Combine Like Terms:
- Now, add all these terms together:
[tex]\[
20x^4 + (-12x^3) + 35x^3 + (-21x^2)
\][/tex]
- Combine the [tex]\(x^3\)[/tex] terms:
[tex]\[
-12x^3 + 35x^3 = 23x^3
\][/tex]
- Thus, the combined polynomial is:
[tex]\[
20x^4 + 23x^3 - 21x^2
\][/tex]
3. Select the Correct Answer:
- From the choices provided, the polynomial that matches our result is Option C: [tex]\(20x^4 + 23x^3 - 21x^2\)[/tex].
Therefore, the correct answer is:
C. [tex]\(20x^4 + 23x^3 - 21x^2\)[/tex]
