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Answer :
Sure! Let's solve the problem step-by-step by multiplying the binomials [tex]\((4x^2 + 7x)\)[/tex] and [tex]\((5x^2 - 3x)\)[/tex].
### Step 1: Use the distributive property (FOIL Method)
First, we will distribute each term in [tex]\((4x^2 + 7x)\)[/tex] by each term in [tex]\((5x^2 - 3x)\)[/tex].
The distributive property of multiplication over addition states that [tex]\( (a + b)(c + d) = ac + ad + bc + bd \)[/tex]. We apply this to each term:
[tex]\[
(4x^2 + 7x)(5x^2 - 3x) = (4x^2 \cdot 5x^2) + (4x^2 \cdot -3x) + (7x \cdot 5x^2) + (7x \cdot -3x)
\][/tex]
### Step 2: Multiply each pair of terms
Perform the multiplication:
1. Multiply [tex]\(4x^2\)[/tex] and [tex]\(5x^2\)[/tex]:
[tex]\[
4x^2 \cdot 5x^2 = 20x^4
\][/tex]
2. Multiply [tex]\(4x^2\)[/tex] and [tex]\(-3x\)[/tex]:
[tex]\[
4x^2 \cdot -3x = -12x^3
\][/tex]
3. Multiply [tex]\(7x\)[/tex] and [tex]\(5x^2\)[/tex]:
[tex]\[
7x \cdot 5x^2 = 35x^3
\][/tex]
4. Multiply [tex]\(7x\)[/tex] and [tex]\(-3x\)[/tex]:
[tex]\[
7x \cdot -3x = -21x^2
\][/tex]
### Step 3: Combine like terms
Now, we add all the terms together:
[tex]\[
20x^4 - 12x^3 + 35x^3 - 21x^2
\][/tex]
Combine the [tex]\(x^3\)[/tex] terms:
[tex]\[
20x^4 + (35x^3 - 12x^3) - 21x^2 = 20x^4 + 23x^3 - 21x^2
\][/tex]
### Step 4: Write the final expression
The final simplified expression is:
[tex]\[
20x^4 + 23x^3 - 21x^2
\][/tex]
### Step 5: Select the correct option
By comparing our result with the given options, we find that option D matches our result. Thus, the correct answer is:
D. [tex]\(20x^4 + 23x^3 - 21x^2\)[/tex]
### Step 1: Use the distributive property (FOIL Method)
First, we will distribute each term in [tex]\((4x^2 + 7x)\)[/tex] by each term in [tex]\((5x^2 - 3x)\)[/tex].
The distributive property of multiplication over addition states that [tex]\( (a + b)(c + d) = ac + ad + bc + bd \)[/tex]. We apply this to each term:
[tex]\[
(4x^2 + 7x)(5x^2 - 3x) = (4x^2 \cdot 5x^2) + (4x^2 \cdot -3x) + (7x \cdot 5x^2) + (7x \cdot -3x)
\][/tex]
### Step 2: Multiply each pair of terms
Perform the multiplication:
1. Multiply [tex]\(4x^2\)[/tex] and [tex]\(5x^2\)[/tex]:
[tex]\[
4x^2 \cdot 5x^2 = 20x^4
\][/tex]
2. Multiply [tex]\(4x^2\)[/tex] and [tex]\(-3x\)[/tex]:
[tex]\[
4x^2 \cdot -3x = -12x^3
\][/tex]
3. Multiply [tex]\(7x\)[/tex] and [tex]\(5x^2\)[/tex]:
[tex]\[
7x \cdot 5x^2 = 35x^3
\][/tex]
4. Multiply [tex]\(7x\)[/tex] and [tex]\(-3x\)[/tex]:
[tex]\[
7x \cdot -3x = -21x^2
\][/tex]
### Step 3: Combine like terms
Now, we add all the terms together:
[tex]\[
20x^4 - 12x^3 + 35x^3 - 21x^2
\][/tex]
Combine the [tex]\(x^3\)[/tex] terms:
[tex]\[
20x^4 + (35x^3 - 12x^3) - 21x^2 = 20x^4 + 23x^3 - 21x^2
\][/tex]
### Step 4: Write the final expression
The final simplified expression is:
[tex]\[
20x^4 + 23x^3 - 21x^2
\][/tex]
### Step 5: Select the correct option
By comparing our result with the given options, we find that option D matches our result. Thus, the correct answer is:
D. [tex]\(20x^4 + 23x^3 - 21x^2\)[/tex]
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