Answer :
Sure! Let's go through a step-by-step process to find which expression is equal to [tex]\((3x - 5)(2x - 7)\)[/tex].
### Step 1: Expand the Expression
To expand the expression [tex]\((3x - 5)(2x - 7)\)[/tex], we need to use distribution, also known as the FOIL (First, Outer, Inner, Last) method. This is how we multiply two binomials.
1. First Terms: Multiply the first terms in each binomial:
- [tex]\(3x \cdot 2x = 6x^2\)[/tex]
2. Outer Terms: Multiply the outer terms in the product:
- [tex]\(3x \cdot (-7) = -21x\)[/tex]
3. Inner Terms: Multiply the inner terms:
- [tex]\(-5 \cdot 2x = -10x\)[/tex]
4. Last Terms: Multiply the last terms in each binomial:
- [tex]\(-5 \cdot (-7) = 35\)[/tex]
### Step 2: Combine Like Terms
After expanding, we combine the terms:
- The [tex]\(x^2\)[/tex] term is: [tex]\(6x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-21x - 10x = -31x\)[/tex]
- The constant term is: [tex]\(35\)[/tex]
Putting it all together, the expanded form of the expression is:
[tex]\[6x^2 - 31x + 35\][/tex]
### Step 3: Match with Given Options
Now, let's compare this expanded expression with the options provided:
- [tex]\(6x^2 + 31x - 35\)[/tex]
- [tex]\(6x^2 - 31x + 35\)[/tex]
- [tex]\(5x^2 - 21x + 12\)[/tex]
- [tex]\(6x^2 - 31x - 12\)[/tex]
The expression we found, [tex]\(6x^2 - 31x + 35\)[/tex], matches exactly with the second option.
Thus, the correct answer is:
[tex]\[6x^2 - 31x + 35\][/tex]
### Step 1: Expand the Expression
To expand the expression [tex]\((3x - 5)(2x - 7)\)[/tex], we need to use distribution, also known as the FOIL (First, Outer, Inner, Last) method. This is how we multiply two binomials.
1. First Terms: Multiply the first terms in each binomial:
- [tex]\(3x \cdot 2x = 6x^2\)[/tex]
2. Outer Terms: Multiply the outer terms in the product:
- [tex]\(3x \cdot (-7) = -21x\)[/tex]
3. Inner Terms: Multiply the inner terms:
- [tex]\(-5 \cdot 2x = -10x\)[/tex]
4. Last Terms: Multiply the last terms in each binomial:
- [tex]\(-5 \cdot (-7) = 35\)[/tex]
### Step 2: Combine Like Terms
After expanding, we combine the terms:
- The [tex]\(x^2\)[/tex] term is: [tex]\(6x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-21x - 10x = -31x\)[/tex]
- The constant term is: [tex]\(35\)[/tex]
Putting it all together, the expanded form of the expression is:
[tex]\[6x^2 - 31x + 35\][/tex]
### Step 3: Match with Given Options
Now, let's compare this expanded expression with the options provided:
- [tex]\(6x^2 + 31x - 35\)[/tex]
- [tex]\(6x^2 - 31x + 35\)[/tex]
- [tex]\(5x^2 - 21x + 12\)[/tex]
- [tex]\(6x^2 - 31x - 12\)[/tex]
The expression we found, [tex]\(6x^2 - 31x + 35\)[/tex], matches exactly with the second option.
Thus, the correct answer is:
[tex]\[6x^2 - 31x + 35\][/tex]
