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Answer :
Sure, let's go through the steps to simplify the expression [tex]\((7x - 3)(4x^2 - 3x - 6)\)[/tex].
### Step-by-Step Solution:
1. Distribute each term in the first binomial by each term in the second trinomial:
[tex]\[
(7x - 3)(4x^2 - 3x - 6)
\][/tex]
This can be broken down into:
[tex]\[
7x \cdot 4x^2 + 7x \cdot (-3x) + 7x \cdot (-6) + (-3) \cdot 4x^2 + (-3) \cdot (-3x) + (-3) \cdot (-6)
\][/tex]
2. Multiply each pair of terms:
- [tex]\(7x \cdot 4x^2 = 28x^3\)[/tex]
- [tex]\(7x \cdot (-3x) = -21x^2\)[/tex]
- [tex]\(7x \cdot (-6) = -42x\)[/tex]
- [tex]\((-3) \cdot 4x^2 = -12x^2\)[/tex]
- [tex]\((-3) \cdot (-3x) = 9x\)[/tex]
- [tex]\((-3) \cdot (-6) = 18\)[/tex]
3. Combine all the results:
[tex]\[
28x^3 - 21x^2 - 42x - 12x^2 + 9x + 18
\][/tex]
4. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-21x^2 - 12x^2 = -33x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-42x + 9x = -33x\)[/tex]
Now, we have:
[tex]\[
28x^3 - 33x^2 - 33x + 18
\][/tex]
So, the correct simplification of [tex]\((7x - 3)(4x^2 - 3x - 6)\)[/tex] is:
[tex]\[
28x^3 - 33x^2 - 33x + 18
\][/tex]
From the available choices, the correct answer is:
4) [tex]\( 28x^3 - 33x^2 - 33x + 18 \)[/tex]
### Step-by-Step Solution:
1. Distribute each term in the first binomial by each term in the second trinomial:
[tex]\[
(7x - 3)(4x^2 - 3x - 6)
\][/tex]
This can be broken down into:
[tex]\[
7x \cdot 4x^2 + 7x \cdot (-3x) + 7x \cdot (-6) + (-3) \cdot 4x^2 + (-3) \cdot (-3x) + (-3) \cdot (-6)
\][/tex]
2. Multiply each pair of terms:
- [tex]\(7x \cdot 4x^2 = 28x^3\)[/tex]
- [tex]\(7x \cdot (-3x) = -21x^2\)[/tex]
- [tex]\(7x \cdot (-6) = -42x\)[/tex]
- [tex]\((-3) \cdot 4x^2 = -12x^2\)[/tex]
- [tex]\((-3) \cdot (-3x) = 9x\)[/tex]
- [tex]\((-3) \cdot (-6) = 18\)[/tex]
3. Combine all the results:
[tex]\[
28x^3 - 21x^2 - 42x - 12x^2 + 9x + 18
\][/tex]
4. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-21x^2 - 12x^2 = -33x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-42x + 9x = -33x\)[/tex]
Now, we have:
[tex]\[
28x^3 - 33x^2 - 33x + 18
\][/tex]
So, the correct simplification of [tex]\((7x - 3)(4x^2 - 3x - 6)\)[/tex] is:
[tex]\[
28x^3 - 33x^2 - 33x + 18
\][/tex]
From the available choices, the correct answer is:
4) [tex]\( 28x^3 - 33x^2 - 33x + 18 \)[/tex]
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