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Answer :
Let's simplify the given expression [tex]\((6x - 5)(2x^2 - 3x - 6)\)[/tex] step-by-step by using the distributive property, also known as the FOIL method when dealing with polynomials.
1. Distribute [tex]\(6x\)[/tex] to each term in the second polynomial:
- [tex]\(6x \times 2x^2 = 12x^3\)[/tex]
- [tex]\(6x \times -3x = -18x^2\)[/tex]
- [tex]\(6x \times -6 = -36x\)[/tex]
2. Distribute [tex]\(-5\)[/tex] to each term in the second polynomial:
- [tex]\(-5 \times 2x^2 = -10x^2\)[/tex]
- [tex]\(-5 \times -3x = 15x\)[/tex]
- [tex]\(-5 \times -6 = 30\)[/tex]
3. Combine all the terms:
[tex]\[
12x^3 - 18x^2 - 36x - 10x^2 + 15x + 30
\][/tex]
4. Combine like terms:
- The [tex]\(x^2\)[/tex] terms: [tex]\(-18x^2 - 10x^2 = -28x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-36x + 15x = -21x\)[/tex]
5. Write down the simplified expression:
[tex]\[
12x^3 - 28x^2 - 21x + 30
\][/tex]
By considering the provided options, the correct simplification is:
[tex]\[12x^3 - 28x^2 - 21x + 30\][/tex]
This matches the option:
[tex]\[ \boxed{12x^3 - 28x^2 - 21x + 30} \][/tex]
1. Distribute [tex]\(6x\)[/tex] to each term in the second polynomial:
- [tex]\(6x \times 2x^2 = 12x^3\)[/tex]
- [tex]\(6x \times -3x = -18x^2\)[/tex]
- [tex]\(6x \times -6 = -36x\)[/tex]
2. Distribute [tex]\(-5\)[/tex] to each term in the second polynomial:
- [tex]\(-5 \times 2x^2 = -10x^2\)[/tex]
- [tex]\(-5 \times -3x = 15x\)[/tex]
- [tex]\(-5 \times -6 = 30\)[/tex]
3. Combine all the terms:
[tex]\[
12x^3 - 18x^2 - 36x - 10x^2 + 15x + 30
\][/tex]
4. Combine like terms:
- The [tex]\(x^2\)[/tex] terms: [tex]\(-18x^2 - 10x^2 = -28x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-36x + 15x = -21x\)[/tex]
5. Write down the simplified expression:
[tex]\[
12x^3 - 28x^2 - 21x + 30
\][/tex]
By considering the provided options, the correct simplification is:
[tex]\[12x^3 - 28x^2 - 21x + 30\][/tex]
This matches the option:
[tex]\[ \boxed{12x^3 - 28x^2 - 21x + 30} \][/tex]
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