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Answer :
To determine which expression is equal to [tex]\((3x - 5)(2x - 7)\)[/tex], we will expand the expression step-by-step and then compare it with the given options.
### Step-by-Step Expansion:
1. Use the distributive property (FOIL method):
- First terms: Multiply the first terms of each binomial: [tex]\(3x \times 2x = 6x^2\)[/tex].
- Outer terms: Multiply the outer terms: [tex]\(3x \times -7 = -21x\)[/tex].
- Inner terms: Multiply the inner terms: [tex]\(-5 \times 2x = -10x\)[/tex].
- Last terms: Multiply the last terms: [tex]\(-5 \times -7 = 35\)[/tex].
2. Combine all the parts:
- Combine all these products: [tex]\(6x^2 - 21x - 10x + 35\)[/tex].
3. Simplify by combining like terms:
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-21x - 10x = -31x\)[/tex].
4. Final expression:
- The expanded expression is [tex]\(6x^2 - 31x + 35\)[/tex].
### Compare with Given Options:
Now, let's compare this expression with the given options:
- [tex]\(6x^2 - 31x - 12\)[/tex]
- [tex]\(5x^2 - 21x + 12\)[/tex]
- [tex]\(6x^2 + 31x - 35\)[/tex]
- [tex]\(6x^2 - 31x + 35\)[/tex]
The expression [tex]\(6x^2 - 31x + 35\)[/tex] matches the expanded form of [tex]\((3x - 5)(2x - 7)\)[/tex].
Therefore, the correct answer is the expression [tex]\(6x^2 - 31x + 35\)[/tex], which corresponds to the fourth option.
### Step-by-Step Expansion:
1. Use the distributive property (FOIL method):
- First terms: Multiply the first terms of each binomial: [tex]\(3x \times 2x = 6x^2\)[/tex].
- Outer terms: Multiply the outer terms: [tex]\(3x \times -7 = -21x\)[/tex].
- Inner terms: Multiply the inner terms: [tex]\(-5 \times 2x = -10x\)[/tex].
- Last terms: Multiply the last terms: [tex]\(-5 \times -7 = 35\)[/tex].
2. Combine all the parts:
- Combine all these products: [tex]\(6x^2 - 21x - 10x + 35\)[/tex].
3. Simplify by combining like terms:
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-21x - 10x = -31x\)[/tex].
4. Final expression:
- The expanded expression is [tex]\(6x^2 - 31x + 35\)[/tex].
### Compare with Given Options:
Now, let's compare this expression with the given options:
- [tex]\(6x^2 - 31x - 12\)[/tex]
- [tex]\(5x^2 - 21x + 12\)[/tex]
- [tex]\(6x^2 + 31x - 35\)[/tex]
- [tex]\(6x^2 - 31x + 35\)[/tex]
The expression [tex]\(6x^2 - 31x + 35\)[/tex] matches the expanded form of [tex]\((3x - 5)(2x - 7)\)[/tex].
Therefore, the correct answer is the expression [tex]\(6x^2 - 31x + 35\)[/tex], which corresponds to the fourth option.
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