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Answer :
Let's solve the expression [tex]\((4x^2 + 7x)(5x^2 - 3x)\)[/tex] by expanding it fully and then combining like terms. We'll work through it step-by-step:
1. Distribute each term from the first polynomial to each term in the second polynomial:
- Multiply the first term in the first polynomial [tex]\(4x^2\)[/tex] by each term in the second polynomial:
- [tex]\(4x^2 \times 5x^2 = 20x^4\)[/tex]
- [tex]\(4x^2 \times (-3x) = -12x^3\)[/tex]
- Multiply the second term in the first polynomial [tex]\(7x\)[/tex] by each term in the second polynomial:
- [tex]\(7x \times 5x^2 = 35x^3\)[/tex]
- [tex]\(7x \times (-3x) = -21x^2\)[/tex]
2. Combine all the terms from the distribution:
[tex]\[
20x^4 - 12x^3 + 35x^3 - 21x^2
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-12x^3 + 35x^3 = 23x^3\)[/tex]
4. Write the simplified expression:
[tex]\[
20x^4 + 23x^3 - 21x^2
\][/tex]
Therefore, the correct answer is D. [tex]\(20x^4 + 23x^3 - 21x^2\)[/tex].
1. Distribute each term from the first polynomial to each term in the second polynomial:
- Multiply the first term in the first polynomial [tex]\(4x^2\)[/tex] by each term in the second polynomial:
- [tex]\(4x^2 \times 5x^2 = 20x^4\)[/tex]
- [tex]\(4x^2 \times (-3x) = -12x^3\)[/tex]
- Multiply the second term in the first polynomial [tex]\(7x\)[/tex] by each term in the second polynomial:
- [tex]\(7x \times 5x^2 = 35x^3\)[/tex]
- [tex]\(7x \times (-3x) = -21x^2\)[/tex]
2. Combine all the terms from the distribution:
[tex]\[
20x^4 - 12x^3 + 35x^3 - 21x^2
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(-12x^3 + 35x^3 = 23x^3\)[/tex]
4. Write the simplified expression:
[tex]\[
20x^4 + 23x^3 - 21x^2
\][/tex]
Therefore, the correct answer is D. [tex]\(20x^4 + 23x^3 - 21x^2\)[/tex].
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