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Answer :
Sure! Let's work through the problem step-by-step to find out which expression is equal to [tex]\((3x - 5)(2x - 7)\)[/tex].
### Step 1: Expand the expression [tex]\((3x - 5)(2x - 7)\)[/tex]
To expand this, we will use the distributive property (also known as the FOIL method for binomials):
1. Multiply the first terms: [tex]\(3x \cdot 2x = 6x^2\)[/tex]
2. Multiply the outer terms: [tex]\(3x \cdot -7 = -21x\)[/tex]
3. Multiply the inner terms: [tex]\(-5 \cdot 2x = -10x\)[/tex]
4. Multiply the last terms: [tex]\(-5 \cdot -7 = 35\)[/tex]
Now add these results together:
[tex]\[6x^2 - 21x - 10x + 35\][/tex]
### Step 2: Combine like terms
Combine the [tex]\(x\)[/tex] terms:
[tex]\[-21x - 10x = -31x\][/tex]
So, the expression becomes:
[tex]\[6x^2 - 31x + 35\][/tex]
### Step 3: Compare with given options
The expression we found is [tex]\(6x^2 - 31x + 35\)[/tex]. Let's compare this with the provided options:
1. [tex]\(6x^2 - 31x + 35\)[/tex]
2. [tex]\(5x^2 - 21x + 12\)[/tex]
3. [tex]\(6x^2 - 31x - 12\)[/tex]
4. [tex]\(6x^2 + 31x - 35\)[/tex]
### Step 4: Identify the correct answer
The expression [tex]\(6x^2 - 31x + 35\)[/tex] matches exactly with Option 1.
Therefore, the correct answer is:
[tex]\[
\boxed{6x^2 - 31x + 35}
\][/tex]
### Step 1: Expand the expression [tex]\((3x - 5)(2x - 7)\)[/tex]
To expand this, we will use the distributive property (also known as the FOIL method for binomials):
1. Multiply the first terms: [tex]\(3x \cdot 2x = 6x^2\)[/tex]
2. Multiply the outer terms: [tex]\(3x \cdot -7 = -21x\)[/tex]
3. Multiply the inner terms: [tex]\(-5 \cdot 2x = -10x\)[/tex]
4. Multiply the last terms: [tex]\(-5 \cdot -7 = 35\)[/tex]
Now add these results together:
[tex]\[6x^2 - 21x - 10x + 35\][/tex]
### Step 2: Combine like terms
Combine the [tex]\(x\)[/tex] terms:
[tex]\[-21x - 10x = -31x\][/tex]
So, the expression becomes:
[tex]\[6x^2 - 31x + 35\][/tex]
### Step 3: Compare with given options
The expression we found is [tex]\(6x^2 - 31x + 35\)[/tex]. Let's compare this with the provided options:
1. [tex]\(6x^2 - 31x + 35\)[/tex]
2. [tex]\(5x^2 - 21x + 12\)[/tex]
3. [tex]\(6x^2 - 31x - 12\)[/tex]
4. [tex]\(6x^2 + 31x - 35\)[/tex]
### Step 4: Identify the correct answer
The expression [tex]\(6x^2 - 31x + 35\)[/tex] matches exactly with Option 1.
Therefore, the correct answer is:
[tex]\[
\boxed{6x^2 - 31x + 35}
\][/tex]
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