Answer :
Sure! Let's simplify the expression [tex]\((6x - 5)(2x^2 - 3x - 6)\)[/tex] step-by-step.
We will distribute each term of the first polynomial [tex]\((6x - 5)\)[/tex] to every term of the second polynomial [tex]\((2x^2 - 3x - 6)\)[/tex]:
1. Distribute [tex]\(6x\)[/tex]:
- [tex]\(6x \cdot 2x^2 = 12x^3\)[/tex]
- [tex]\(6x \cdot (-3x) = -18x^2\)[/tex]
- [tex]\(6x \cdot (-6) = -36x\)[/tex]
2. Distribute [tex]\(-5\)[/tex]:
- [tex]\(-5 \cdot 2x^2 = -10x^2\)[/tex]
- [tex]\(-5 \cdot (-3x) = 15x\)[/tex]
- [tex]\(-5 \cdot (-6) = 30\)[/tex]
Now, we combine all of these terms together:
[tex]\[ 12x^3 - 18x^2 - 36x - 10x^2 + 15x + 30 \][/tex]
Next, we combine like terms:
- The [tex]\(x^3\)[/tex] term: [tex]\(12x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(-18x^2 - 10x^2 = -28x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-36x + 15x = -21x\)[/tex]
- The constant term: [tex]\(+ 30\)[/tex]
So, the simplified expression is:
[tex]\[ 12x^3 - 28x^2 - 21x + 30 \][/tex]
Therefore, 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]
We will distribute each term of the first polynomial [tex]\((6x - 5)\)[/tex] to every term of the second polynomial [tex]\((2x^2 - 3x - 6)\)[/tex]:
1. Distribute [tex]\(6x\)[/tex]:
- [tex]\(6x \cdot 2x^2 = 12x^3\)[/tex]
- [tex]\(6x \cdot (-3x) = -18x^2\)[/tex]
- [tex]\(6x \cdot (-6) = -36x\)[/tex]
2. Distribute [tex]\(-5\)[/tex]:
- [tex]\(-5 \cdot 2x^2 = -10x^2\)[/tex]
- [tex]\(-5 \cdot (-3x) = 15x\)[/tex]
- [tex]\(-5 \cdot (-6) = 30\)[/tex]
Now, we combine all of these terms together:
[tex]\[ 12x^3 - 18x^2 - 36x - 10x^2 + 15x + 30 \][/tex]
Next, we combine like terms:
- The [tex]\(x^3\)[/tex] term: [tex]\(12x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(-18x^2 - 10x^2 = -28x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-36x + 15x = -21x\)[/tex]
- The constant term: [tex]\(+ 30\)[/tex]
So, the simplified expression is:
[tex]\[ 12x^3 - 28x^2 - 21x + 30 \][/tex]
Therefore, 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]
