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

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

Answer :

Sure, let's solve the expression [tex]\((3x - 5)(2x - 7)\)[/tex].

### Step-by-step Solution

1. Distribute each term in the first polynomial [tex]\((3x - 5)\)[/tex] to each term in the second polynomial [tex]\((2x - 7)\)[/tex]:

[tex]\[
(3x - 5)(2x - 7) = 3x \cdot 2x + 3x \cdot (-7) + (-5) \cdot 2x + (-5) \cdot (-7)
\][/tex]

2. Multiply the terms:

- [tex]\(3x \cdot 2x = 6x^2\)[/tex]
- [tex]\(3x \cdot (-7) = -21x\)[/tex]
- [tex]\((-5) \cdot 2x = -10x\)[/tex]
- [tex]\((-5) \cdot (-7) = 35\)[/tex]

3. Combine all these results:

[tex]\[
6x^2 - 21x - 10x + 35
\][/tex]

4. Combine the like terms (the [tex]\(x\)[/tex] terms):

[tex]\[
-21x - 10x = -31x
\][/tex]

5. Write the final expression:

[tex]\[
6x^2 - 31x + 35
\][/tex]

So, the expression [tex]\((3x - 5)(2x - 7)\)[/tex] simplifies to:

[tex]\[6x^2 - 31x + 35\][/tex]

Therefore, the correct answer is:

[tex]\[ \boxed{6x^2 - 31x + 35} \][/tex]

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