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

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

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

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

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

Answer :

Sure! To determine which expression is equal to [tex]\((3x - 5)(2x - 7)\)[/tex], we can use the distributive property, commonly known as the FOIL method for multiplying binomials. Here are the steps broken down:

1. First Terms:
Multiply the first terms in each binomial:
[tex]\[
3x \times 2x = 6x^2
\][/tex]

2. Outer Terms:
Multiply the outer terms:
[tex]\[
3x \times (-7) = -21x
\][/tex]

3. Inner Terms:
Multiply the inner terms:
[tex]\[
-5 \times 2x = -10x
\][/tex]

4. Last Terms:
Multiply the last terms:
[tex]\[
-5 \times (-7) = 35
\][/tex]

Now, let's combine all these results into a single expression:

- The result from the first terms is [tex]\(6x^2\)[/tex].
- The results from the outer and inner terms are [tex]\(-21x\)[/tex] and [tex]\(-10x\)[/tex], which need to be combined:
[tex]\[
-21x - 10x = -31x
\][/tex]
- Lastly, the result from the last terms is [tex]\(35\)[/tex].

Putting it all together, the expression is:

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

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

This matches the choice: [tex]\(6x^2 - 31x + 35\)[/tex].

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