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Answer :
Let's solve the expression [tex]\((4x^2 + 7x)(5x^2 - 3x)\)[/tex] by expanding it using the distributive property (also known as FOIL for binomials):
1. Multiply each term in the first expression by each term in the second expression:
- First, multiply [tex]\(4x^2\)[/tex] by [tex]\(5x^2\)[/tex]:
[tex]\[ 4x^2 \cdot 5x^2 = 20x^4 \][/tex]
- Next, multiply [tex]\(4x^2\)[/tex] by [tex]\(-3x\)[/tex]:
[tex]\[ 4x^2 \cdot (-3x) = -12x^3 \][/tex]
- Then, multiply [tex]\(7x\)[/tex] by [tex]\(5x^2\)[/tex]:
[tex]\[ 7x \cdot 5x^2 = 35x^3 \][/tex]
- Finally, multiply [tex]\(7x\)[/tex] by [tex]\(-3x\)[/tex]:
[tex]\[ 7x \cdot (-3x) = -21x^2 \][/tex]
2. Combine the results:
Now, we'll add up all these terms:
[tex]\[ 20x^4 - 12x^3 + 35x^3 - 21x^2 \][/tex]
3. Combine like terms:
The like terms here are the [tex]\(x^3\)[/tex] terms: [tex]\(-12x^3\)[/tex] and [tex]\(35x^3\)[/tex]. Combine them:
[tex]\[ -12x^3 + 35x^3 = 23x^3 \][/tex]
4. Write the final expanded expression:
So, the expanded expression is:
[tex]\[ 20x^4 + 23x^3 - 21x^2 \][/tex]
Thus, the correct answer corresponds to option C: [tex]\(20x^4 + 23x^3 - 21x^2\)[/tex].
1. Multiply each term in the first expression by each term in the second expression:
- First, multiply [tex]\(4x^2\)[/tex] by [tex]\(5x^2\)[/tex]:
[tex]\[ 4x^2 \cdot 5x^2 = 20x^4 \][/tex]
- Next, multiply [tex]\(4x^2\)[/tex] by [tex]\(-3x\)[/tex]:
[tex]\[ 4x^2 \cdot (-3x) = -12x^3 \][/tex]
- Then, multiply [tex]\(7x\)[/tex] by [tex]\(5x^2\)[/tex]:
[tex]\[ 7x \cdot 5x^2 = 35x^3 \][/tex]
- Finally, multiply [tex]\(7x\)[/tex] by [tex]\(-3x\)[/tex]:
[tex]\[ 7x \cdot (-3x) = -21x^2 \][/tex]
2. Combine the results:
Now, we'll add up all these terms:
[tex]\[ 20x^4 - 12x^3 + 35x^3 - 21x^2 \][/tex]
3. Combine like terms:
The like terms here are the [tex]\(x^3\)[/tex] terms: [tex]\(-12x^3\)[/tex] and [tex]\(35x^3\)[/tex]. Combine them:
[tex]\[ -12x^3 + 35x^3 = 23x^3 \][/tex]
4. Write the final expanded expression:
So, the expanded expression is:
[tex]\[ 20x^4 + 23x^3 - 21x^2 \][/tex]
Thus, the correct answer corresponds to option C: [tex]\(20x^4 + 23x^3 - 21x^2\)[/tex].
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