Answer :
Sure, let’s solve the expression [tex]\((3x - 5)(2x - 7)\)[/tex] step-by-step to see which of the given expressions it matches.
We start by expanding the product of the two binomials:
1. [tex]\((3x - 5)(2x - 7)\)[/tex]
To expand this, we use the distributive property (also known as FOIL method for binomials):
- First, multiply the first terms in each binomial: [tex]\(3x \cdot 2x = 6x^2\)[/tex]
- Then, multiply the outer terms: [tex]\(3x \cdot (-7) = -21x\)[/tex]
- Next, multiply the inner terms: [tex]\(-5 \cdot 2x = -10x\)[/tex]
- Finally, multiply the last terms: [tex]\(-5 \cdot (-7) = 35\)[/tex]
Now we add all these products together:
[tex]\[ 6x^2 + (-21x) + (-10x) + 35 \][/tex]
Combine the like terms:
[tex]\[ 6x^2 - 21x - 10x + 35 \][/tex]
[tex]\[ 6x^2 - 31x + 35 \][/tex]
Therefore, the expanded form of [tex]\((3x - 5)(2x - 7)\)[/tex] is [tex]\(6x^2 - 31x + 35\)[/tex].
Comparing this to the given options:
[tex]\[
\begin{array}{l}
6x^2 + 31x - 35 \\
6x^2 - 31x + 35 \\
6x^2 - 31x - 12 \\
5x^2 - 21x + 12 \\
\end{array}
\][/tex]
We can see that [tex]\(6x^2 - 31x + 35\)[/tex] matches the second option.
So, the correct answer is:
[tex]\[ 6x^2 - 31x + 35 \][/tex]
We start by expanding the product of the two binomials:
1. [tex]\((3x - 5)(2x - 7)\)[/tex]
To expand this, we use the distributive property (also known as FOIL method for binomials):
- First, multiply the first terms in each binomial: [tex]\(3x \cdot 2x = 6x^2\)[/tex]
- Then, multiply the outer terms: [tex]\(3x \cdot (-7) = -21x\)[/tex]
- Next, multiply the inner terms: [tex]\(-5 \cdot 2x = -10x\)[/tex]
- Finally, multiply the last terms: [tex]\(-5 \cdot (-7) = 35\)[/tex]
Now we add all these products together:
[tex]\[ 6x^2 + (-21x) + (-10x) + 35 \][/tex]
Combine the like terms:
[tex]\[ 6x^2 - 21x - 10x + 35 \][/tex]
[tex]\[ 6x^2 - 31x + 35 \][/tex]
Therefore, the expanded form of [tex]\((3x - 5)(2x - 7)\)[/tex] is [tex]\(6x^2 - 31x + 35\)[/tex].
Comparing this to the given options:
[tex]\[
\begin{array}{l}
6x^2 + 31x - 35 \\
6x^2 - 31x + 35 \\
6x^2 - 31x - 12 \\
5x^2 - 21x + 12 \\
\end{array}
\][/tex]
We can see that [tex]\(6x^2 - 31x + 35\)[/tex] matches the second option.
So, the correct answer is:
[tex]\[ 6x^2 - 31x + 35 \][/tex]
