Of course! Let's break down the expression step by step:
We want to simplify the expression:
[tex]\[
\frac{a^2 b^2 + b^2 c^2}{a b - b c}
\][/tex]
### Step 1: Factor the Numerator
Notice that the numerator [tex]\(a^2 b^2 + b^2 c^2\)[/tex] has a common factor of [tex]\(b^2\)[/tex]. So, we can factor [tex]\(b^2\)[/tex] out of the numerator:
[tex]\[
a^2 b^2 + b^2 c^2 = b^2 (a^2 + c^2)
\][/tex]
Now our expression becomes:
[tex]\[
\frac{b^2 (a^2 + c^2)}{a b - b c}
\][/tex]
### Step 2: Factor the Denominator
Similarly, the denominator [tex]\(a b - b c\)[/tex] has a common factor of [tex]\(b\)[/tex]. We can factor [tex]\(b\)[/tex] out of the denominator:
[tex]\[
a b - b c = b (a - c)
\][/tex]
Now our expression looks like:
[tex]\[
\frac{b^2 (a^2 + c^2)}{b (a - c)}
\][/tex]
### Step 3: Simplify the Expression
Notice that both the numerator and the denominator contain the factor [tex]\(b\)[/tex]. We can cancel one [tex]\(b\)[/tex] from the numerator and the denominator:
[tex]\[
\frac{b^2 (a^2 + c^2)}{b (a - c)} = \frac{b (a^2 + c^2)}{a - c}
\][/tex]
### Final Simplified Expression
So, the expression [tex]\(\frac{a^2 b^2 + b^2 c^2}{a b - b c}\)[/tex] simplifies to:
[tex]\[
\frac{b (a^2 + c^2)}{a - c}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{\frac{b (a^2 + c^2)}{a - c}}
\][/tex]
