College

Which expression is equal to [tex]$(3x-5)(2x-7)$[/tex]?

A. [tex]6x^2 + 31x - 35[/tex]
B. [tex]6x^2 - 31x - 12[/tex]
C. [tex]5x^2 - 21x + 12[/tex]
D. [tex]6x^2 - 31x + 35[/tex]

Answer :

Let's verify which expression is equal to [tex]\((3x - 5)(2x - 7)\)[/tex] by expanding the given expression step-by-step.

### Step-by-Step Solution:

1. Distribute Each Term in the Expression:
- Use the distributive property (also known as the FOIL method for binomials). FOIL stands for First, Outer, Inner, Last.

[tex]\[
(3x - 5)(2x - 7) = (3x) \cdot (2x) \quad \text{(First terms)} \\
+ (3x) \cdot (-7) \quad \text{(Outer terms)} \\
+ (-5) \cdot (2x) \quad \text{(Inner terms)} \\
+ (-5) \cdot (-7) \quad \text{(Last terms)}
\][/tex]

2. Multiply Each Pair of 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:
- Combine the terms with [tex]\(x\)[/tex]:
[tex]\[
-21x + (-10x) = -31x
\][/tex]

4. Form the Final Expression:
- When we add all the terms together, we get:
[tex]\[
6x^2 - 31x + 35
\][/tex]

### Conclusion:
After expanding and simplifying the expression, we found that:

[tex]\[
(3x - 5)(2x - 7) = 6x^2 - 31x + 35
\][/tex]

Thus, the correct expression is:
[tex]\[
6x^2 - 31x + 35
\][/tex]

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