Let's simplify the given expression step by step:
[tex]\[
= 2 + \frac{\frac{b^4 - 4a^2c^2}{a^4}}{\frac{c^2}{a^2}} - \frac{2c^2}{a^2}
\][/tex]
1. Simplify the fraction [tex]\(\frac{\frac{b^4 - 4a^2c^2}{a^4}}{\frac{c^2}{a^2}}\)[/tex]:
First, simplify the overall fraction:
[tex]\[
\frac{\frac{b^4 - 4a^2c^2}{a^4}}{\frac{c^2}{a^2}} = \frac{b^4 - 4a^2c^2}{a^4} \cdot \frac{a^2}{c^2}
\][/tex]
2. Combine and simplify:
[tex]\[
= \frac{(b^4 - 4a^2c^2) \cdot a^2}{a^4 \cdot c^2}
\][/tex]
Simplify the expression by combining the [tex]\( a^2 \)[/tex] terms:
[tex]\[
= \frac{b^4 - 4a^2c^2}{a^2c^2}
\][/tex]
3. Combine this result back into the original expression:
Now we have the simplified middle term:
[tex]\[
= 2 + \frac{b^4 - 4a^2c^2}{a^2c^2} - \frac{2c^2}{a^2}
\][/tex]
4. Distribute the fraction in the middle term:
Separate the fraction:
[tex]\[
= 2 + \frac{b^4}{a^2c^2} - \frac{4a^2c^2}{a^2c^2} - \frac{2c^2}{a^2}
\][/tex]
5. Simplify the fraction [tex]\(\frac{4a^2c^2}{a^2c^2}\)[/tex]:
[tex]\[
= \frac{4a^2c^2}{a^2c^2} = 4
\][/tex]
Plugging this back into the expression:
[tex]\[
= 2 + \frac{b^4}{a^2c^2} - 4 - \frac{2c^2}{a^2}
\][/tex]
6. Combine the constants and simplify:
[tex]\[
= (2 - 4) + \frac{b^4}{a^2c^2} - \frac{2c^2}{a^2}
\][/tex]
[tex]\[
= -2 + \frac{b^4}{a^2c^2} - \frac{2c^2}{a^2}
\][/tex]
The final simplified expression is:
[tex]\[
-2 + \frac{b^4}{a^2c^2} - \frac{2c^2}{a^2}
\][/tex]
This is the simplified form of the given expression.
