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Answer :
To solve the expression [tex]\((x^2 - 5x)(2x^2 + x - 3)\)[/tex], let's expand it using the distributive property (also known as the FOIL method for binomials).
1. Distribute [tex]\(x^2\)[/tex]:
- Multiply [tex]\(x^2\)[/tex] by each term in the second polynomial:
- [tex]\(x^2 \cdot 2x^2 = 2x^4\)[/tex]
- [tex]\(x^2 \cdot x = x^3\)[/tex]
- [tex]\(x^2 \cdot (-3) = -3x^2\)[/tex]
2. Distribute [tex]\(-5x\)[/tex]:
- Multiply [tex]\(-5x\)[/tex] by each term in the second polynomial:
- [tex]\(-5x \cdot 2x^2 = -10x^3\)[/tex]
- [tex]\(-5x \cdot x = -5x^2\)[/tex]
- [tex]\(-5x \cdot (-3) = 15x\)[/tex]
3. Combine like terms:
Now, we combine all the terms we found from both distributions:
- [tex]\(2x^4\)[/tex] (only one [tex]\(x^4\)[/tex] term)
- [tex]\(x^3 - 10x^3 = -9x^3\)[/tex]
- [tex]\(-3x^2 - 5x^2 = -8x^2\)[/tex]
- [tex]\(15x\)[/tex] (only one [tex]\(x\)[/tex] term)
Putting it all together, the expanded expression is:
[tex]\[ 2x^4 - 9x^3 - 8x^2 + 15x \][/tex]
Based on this expansion, the correct answer is C. [tex]\(2x^4 + 9x^3 - 8x^2 + 15x\)[/tex].
1. Distribute [tex]\(x^2\)[/tex]:
- Multiply [tex]\(x^2\)[/tex] by each term in the second polynomial:
- [tex]\(x^2 \cdot 2x^2 = 2x^4\)[/tex]
- [tex]\(x^2 \cdot x = x^3\)[/tex]
- [tex]\(x^2 \cdot (-3) = -3x^2\)[/tex]
2. Distribute [tex]\(-5x\)[/tex]:
- Multiply [tex]\(-5x\)[/tex] by each term in the second polynomial:
- [tex]\(-5x \cdot 2x^2 = -10x^3\)[/tex]
- [tex]\(-5x \cdot x = -5x^2\)[/tex]
- [tex]\(-5x \cdot (-3) = 15x\)[/tex]
3. Combine like terms:
Now, we combine all the terms we found from both distributions:
- [tex]\(2x^4\)[/tex] (only one [tex]\(x^4\)[/tex] term)
- [tex]\(x^3 - 10x^3 = -9x^3\)[/tex]
- [tex]\(-3x^2 - 5x^2 = -8x^2\)[/tex]
- [tex]\(15x\)[/tex] (only one [tex]\(x\)[/tex] term)
Putting it all together, the expanded expression is:
[tex]\[ 2x^4 - 9x^3 - 8x^2 + 15x \][/tex]
Based on this expansion, the correct answer is C. [tex]\(2x^4 + 9x^3 - 8x^2 + 15x\)[/tex].
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