Answer :
Sure! Let's solve the problem of multiplying the polynomials [tex]\((4x^2 + 7x)\)[/tex] and [tex]\((5x^2 - 3x)\)[/tex] step by step.
### Step 1: Use the distributive property
To multiply these polynomials, use the distributive property (also known as the FOIL method for binomials), applying each term from the first polynomial to each term of the second polynomial.
1. Multiply [tex]\(4x^2\)[/tex] by [tex]\(5x^2\)[/tex]:
[tex]\[
4x^2 \times 5x^2 = 20x^4
\][/tex]
2. Multiply [tex]\(4x^2\)[/tex] by [tex]\(-3x\)[/tex]:
[tex]\[
4x^2 \times (-3x) = -12x^3
\][/tex]
3. Multiply [tex]\(7x\)[/tex] by [tex]\(5x^2\)[/tex]:
[tex]\[
7x \times 5x^2 = 35x^3
\][/tex]
4. Multiply [tex]\(7x\)[/tex] by [tex]\(-3x\)[/tex]:
[tex]\[
7x \times (-3x) = -21x^2
\][/tex]
### Step 2: Combine like terms
Now sum all the results from the multiplications:
[tex]\[
20x^4 + (-12x^3) + 35x^3 + (-21x^2)
\][/tex]
Combine like terms:
- The [tex]\(x^3\)[/tex] terms: [tex]\((-12x^3 + 35x^3 = 23x^3)\)[/tex]
Putting it all together:
[tex]\[
20x^4 + 23x^3 - 21x^2
\][/tex]
### Conclusion
After simplifying, the product of the polynomials is:
[tex]\[
20x^4 + 23x^3 - 21x^2
\][/tex]
So, the correct answer is C. [tex]\(20x^4 + 23x^3 - 21x^2\)[/tex].
### Step 1: Use the distributive property
To multiply these polynomials, use the distributive property (also known as the FOIL method for binomials), applying each term from the first polynomial to each term of the second polynomial.
1. Multiply [tex]\(4x^2\)[/tex] by [tex]\(5x^2\)[/tex]:
[tex]\[
4x^2 \times 5x^2 = 20x^4
\][/tex]
2. Multiply [tex]\(4x^2\)[/tex] by [tex]\(-3x\)[/tex]:
[tex]\[
4x^2 \times (-3x) = -12x^3
\][/tex]
3. Multiply [tex]\(7x\)[/tex] by [tex]\(5x^2\)[/tex]:
[tex]\[
7x \times 5x^2 = 35x^3
\][/tex]
4. Multiply [tex]\(7x\)[/tex] by [tex]\(-3x\)[/tex]:
[tex]\[
7x \times (-3x) = -21x^2
\][/tex]
### Step 2: Combine like terms
Now sum all the results from the multiplications:
[tex]\[
20x^4 + (-12x^3) + 35x^3 + (-21x^2)
\][/tex]
Combine like terms:
- The [tex]\(x^3\)[/tex] terms: [tex]\((-12x^3 + 35x^3 = 23x^3)\)[/tex]
Putting it all together:
[tex]\[
20x^4 + 23x^3 - 21x^2
\][/tex]
### Conclusion
After simplifying, the product of the polynomials is:
[tex]\[
20x^4 + 23x^3 - 21x^2
\][/tex]
So, the correct answer is C. [tex]\(20x^4 + 23x^3 - 21x^2\)[/tex].
