To solve the given expression [tex]\(\frac{45k^2 - 90k}{54k}\)[/tex], we need to simplify it step-by-step. Let's go through the process:
1. Factor the numerator:
[tex]\[
45k^2 - 90k
\][/tex]
You can factor out the greatest common factor (GCF) from the terms in the numerator. The GCF of [tex]\(45k^2\)[/tex] and [tex]\(90k\)[/tex] is [tex]\(45k\)[/tex]. So we factor out [tex]\(45k\)[/tex]:
[tex]\[
45k(k - 2)
\][/tex]
2. Rewrite the expression with the factored numerator:
[tex]\[
\frac{45k(k - 2)}{54k}
\][/tex]
3. Simplify the fraction:
First, look for common factors in the numerator and the denominator. We see that both the numerator [tex]\(45k(k - 2)\)[/tex] and the denominator [tex]\(54k\)[/tex] have a common factor of [tex]\(9k\)[/tex].
Factor out [tex]\(9k\)[/tex] from both the numerator and the denominator:
[tex]\[
\frac{45k(k - 2)}{54k} = \frac{(5k)(9)(k - 2)}{6(9)k}
\][/tex]
4. Simplify further:
[tex]\[
\frac{(5k)(9)(k - 2)}{6(9)k} = \frac{5(k - 2)}{6}
\][/tex]
Here, the [tex]\(9\)[/tex] and [tex]\(k\)[/tex] in both the numerator and denominator cancel each other out.
5. Final expression:
[tex]\[
\frac{5(k - 2)}{6}
\][/tex]
Thus, the simplified form of the given expression [tex]\(\frac{45k^2 - 90k}{54k}\)[/tex] is [tex]\(\frac{5(k - 2)}{6}\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{B) \frac{5(k-2)}{6}}
\][/tex]
