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

A. [tex]$6x^2 - 31x - 12$[/tex]

B. [tex]$5x^2 - 21x + 12$[/tex]

C. [tex]$6x^2 - 31x + 35$[/tex]

D. [tex]$6x^2 + 31x - 35$[/tex]

To determine which expression is equal to [tex]$(3x - 5)(2x - 7)$[/tex], we can expand the given product:

1. Distribute each term in the first binomial to each term in the second binomial:
- First, distribute [tex]$3x$[/tex] to both terms in [tex]$2x - 7$[/tex]:
[tex]\[
3x \cdot 2x = 6x^2
\][/tex]
[tex]\[
3x \cdot (-7) = -21x
\][/tex]

- Next, distribute [tex]$-5$[/tex] to both terms in [tex]$2x - 7$[/tex]:
[tex]\[
-5 \cdot 2x = -10x
\][/tex]
[tex]\[
-5 \cdot (-7) = 35
\][/tex]

2. Combine all of these products:
[tex]\[
6x^2 - 21x - 10x + 35
\][/tex]

3. Combine like terms:
[tex]\[
6x^2 - 31x + 35
\][/tex]

So, the expanded expression is [tex]$6x^2 - 31x + 35$[/tex].

Let's match this expression with the given options:

1. [tex]$6x^2 - 31x - 12$[/tex]
2. [tex]$5x^2 - 21x + 12$[/tex]
3. [tex]$6x^2 - 31x + 35$[/tex]
4. [tex]$6x^2 + 31x - 35$[/tex]

The expression [tex]$6x^2 - 31x + 35$[/tex] matches exactly with the third option.

Therefore, the expression that is equal to [tex]$(3x - 5)(2x - 7)$[/tex] is:
[tex]\[ \boxed{6x^2 - 31x + 35} \][/tex]