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]
### 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]
