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Answer :
Sure! Let's break down the problem and find which expression is equal to [tex]\((3x - 5)(2x - 7)\)[/tex].
### Step-by-Step Solution:
1. Distribute to Multiply the Binomials:
To find the product of [tex]\((3x - 5)(2x - 7)\)[/tex], we'll use the distributive property (also known as the FOIL method for binomials):
[tex]\[
(3x - 5)(2x - 7)
\][/tex]
We need to multiply each term in the first binomial by each term in the second binomial.
2. First, Inner, Outer, and Last:
Let's multiply the terms:
- First terms: [tex]\(3x \cdot 2x = 6x^2\)[/tex]
- Outer terms: [tex]\(3x \cdot (-7) = -21x\)[/tex]
- Inner terms: [tex]\((-5) \cdot 2x = -10x\)[/tex]
- Last terms: [tex]\((-5) \cdot (-7) = 35\)[/tex]
3. Combine Like Terms:
Now, we add all these results together:
[tex]\[
6x^2 - 21x - 10x + 35
\][/tex]
Combine the [tex]\(x\)[/tex]-terms:
[tex]\[
6x^2 - 31x + 35
\][/tex]
4. Identify the Correct Expression:
The expression we obtained is:
[tex]\[
6x^2 - 31x + 35
\][/tex]
Now, let's compare this with the given choices:
- [tex]\(6x^2 + 31x - 35\)[/tex]
- [tex]\(6x^2 - 31x - 12\)[/tex]
- [tex]\(5x^2 - 21x + 12\)[/tex]
- [tex]\(6x^2 - 31x + 35\)[/tex]
The expression that matches is:
[tex]\[
6x^2 - 31x + 35
\][/tex]
### Conclusion:
The expression [tex]\((3x - 5)(2x - 7)\)[/tex] is equal to:
[tex]\[
6x^2 - 31x + 35
\][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{6x^2 - 31x + 35} \][/tex]
### Step-by-Step Solution:
1. Distribute to Multiply the Binomials:
To find the product of [tex]\((3x - 5)(2x - 7)\)[/tex], we'll use the distributive property (also known as the FOIL method for binomials):
[tex]\[
(3x - 5)(2x - 7)
\][/tex]
We need to multiply each term in the first binomial by each term in the second binomial.
2. First, Inner, Outer, and Last:
Let's multiply the terms:
- First terms: [tex]\(3x \cdot 2x = 6x^2\)[/tex]
- Outer terms: [tex]\(3x \cdot (-7) = -21x\)[/tex]
- Inner terms: [tex]\((-5) \cdot 2x = -10x\)[/tex]
- Last terms: [tex]\((-5) \cdot (-7) = 35\)[/tex]
3. Combine Like Terms:
Now, we add all these results together:
[tex]\[
6x^2 - 21x - 10x + 35
\][/tex]
Combine the [tex]\(x\)[/tex]-terms:
[tex]\[
6x^2 - 31x + 35
\][/tex]
4. Identify the Correct Expression:
The expression we obtained is:
[tex]\[
6x^2 - 31x + 35
\][/tex]
Now, let's compare this with the given choices:
- [tex]\(6x^2 + 31x - 35\)[/tex]
- [tex]\(6x^2 - 31x - 12\)[/tex]
- [tex]\(5x^2 - 21x + 12\)[/tex]
- [tex]\(6x^2 - 31x + 35\)[/tex]
The expression that matches is:
[tex]\[
6x^2 - 31x + 35
\][/tex]
### Conclusion:
The expression [tex]\((3x - 5)(2x - 7)\)[/tex] is equal to:
[tex]\[
6x^2 - 31x + 35
\][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{6x^2 - 31x + 35} \][/tex]
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