Answer :
Sure! Let's go through the steps to find which expression is equal to [tex]\((3x - 5)(2x - 7)\)[/tex].
1. Distribute Each Term:
First, we distribute each term in the first binomial ([tex]\(3x - 5\)[/tex]) by each term in the second binomial ([tex]\(2x - 7\)[/tex]):
[tex]\[
(3x - 5)(2x - 7) = 3x(2x) + 3x(-7) + (-5)(2x) + (-5)(-7)
\][/tex]
2. Multiply Each Pair of 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 Like Terms:
Now we add all these results together:
[tex]\[
6x^2 + (-21x) + (-10x) + 35
\][/tex]
Combine the [tex]\(x\)[/tex]-terms ([tex]\(-21x\)[/tex] and [tex]\(-10x\)[/tex]):
[tex]\[
6x^2 - 21x - 10x + 35 = 6x^2 - 31x + 35
\][/tex]
4. Final Expression:
After combining like terms, we get the final expanded expression:
[tex]\[
6x^2 - 31x + 35
\][/tex]
So, the expression that is equal to [tex]\((3x - 5)(2x - 7)\)[/tex] is:
[tex]\[6x^2 - 31x + 35\][/tex]
Hence, the correct answer is:
[tex]\[ 6x^2 - 31x + 35 \][/tex]
1. Distribute Each Term:
First, we distribute each term in the first binomial ([tex]\(3x - 5\)[/tex]) by each term in the second binomial ([tex]\(2x - 7\)[/tex]):
[tex]\[
(3x - 5)(2x - 7) = 3x(2x) + 3x(-7) + (-5)(2x) + (-5)(-7)
\][/tex]
2. Multiply Each Pair of 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 Like Terms:
Now we add all these results together:
[tex]\[
6x^2 + (-21x) + (-10x) + 35
\][/tex]
Combine the [tex]\(x\)[/tex]-terms ([tex]\(-21x\)[/tex] and [tex]\(-10x\)[/tex]):
[tex]\[
6x^2 - 21x - 10x + 35 = 6x^2 - 31x + 35
\][/tex]
4. Final Expression:
After combining like terms, we get the final expanded expression:
[tex]\[
6x^2 - 31x + 35
\][/tex]
So, the expression that is equal to [tex]\((3x - 5)(2x - 7)\)[/tex] is:
[tex]\[6x^2 - 31x + 35\][/tex]
Hence, the correct answer is:
[tex]\[ 6x^2 - 31x + 35 \][/tex]
