College

Multiply:

[tex] \left(4x^2 + 7x\right)\left(5x^2 - 3x\right) [/tex]

A. [tex] 20x^4 + 35x^2 - 21x [/tex]
B. [tex] 20x^4 + 23x^3 - 21x^2 [/tex]
C. [tex] 20x^4 + 23x^2 - 21x [/tex]
D. [tex] 20x^4 + 35x^3 - 21x^2 [/tex]

Answer :

Sure! Let's multiply the two polynomials step by step:

[tex]\[
(4x^2 + 7x)(5x^2 - 3x)
\][/tex]

We will use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial.

### Step 1: Multiply each term in the first polynomial by each term in the second polynomial

1. Multiply [tex]\(4x^2\)[/tex] by [tex]\(5x^2\)[/tex]:
[tex]\[
4x^2 \cdot 5x^2 = 20x^4
\][/tex]

2. Multiply [tex]\(4x^2\)[/tex] by [tex]\(-3x\)[/tex]:
[tex]\[
4x^2 \cdot (-3x) = -12x^3
\][/tex]

3. Multiply [tex]\(7x\)[/tex] by [tex]\(5x^2\)[/tex]:
[tex]\[
7x \cdot 5x^2 = 35x^3
\][/tex]

4. Multiply [tex]\(7x\)[/tex] by [tex]\(-3x\)[/tex]:
[tex]\[
7x \cdot (-3x) = -21x^2
\][/tex]

### Step 2: Combine all the products

[tex]\[
20x^4 - 12x^3 + 35x^3 - 21x^2
\][/tex]

### Step 3: Combine like terms

Combine the [tex]\(x^3\)[/tex] terms:
[tex]\[
-12x^3 + 35x^3 = 23x^3
\][/tex]

So our final expression is:
[tex]\[
20x^4 + 23x^3 - 21x^2
\][/tex]

### Conclusion

The correct answer is:

B. [tex]\(20x^4 + 23x^3 - 21x^2\)[/tex]

This matches the given choices.