The equation you provided in the image is as follows:

$\left( \frac{\frac{7}{x^4} - \frac{3}{x^4} + x \cdot x^4}{\frac{1}{x^4}} \right)^2 = x \left(x^2 + x - 1\right)^2$

Let’s simplify the left-hand side (LHS) step by step:

Step 1: Simplify the numerator of the fraction inside the parentheses:

The numerator is:

$\frac{7}{x^4} - \frac{3}{x^4} + x \cdot x^4$

$= \frac{7 - 3}{x^4} + x \cdot x^4 = \frac{4}{x^4} + x^5$

Step 2: The whole expression inside the parentheses becomes:

$\frac{\frac{4}{x^4} + x^5}{\frac{1}{x^4}}$

Step 3: Simplify the division:

Dividing the numerator by the denominator means multiplying the numerator by the reciprocal of the denominator:

$\left(\frac{4}{x^4} + x^5\right) \cdot x^4 = 4 + x^9$

Step 4: Square the simplified expression:

Now we square the expression:

$(4 + x^9)^2$

Thus, the LHS becomes:

$(4 + x^9)^2$

Step 5: Compare with the right-hand side (RHS):

The RHS is $x(x^2 + x - 1)^2$.

So, the equation now reads:

$(4 + x^9)^2 = x(x^2 + x - 1)^2$

This is the simplified form of the given expression. Both sides are squared terms, and you can further explore by expanding them if needed or by solving for specific values of $x$.

Would you like me to continue with further expansion or solving? Feel free to ask!

Related Questions:

1. How do you expand $(4 + x^9)^2$?
2. Can this equation be solved for specific values of $x$?
3. How does the simplification of complex fractions work?
4. What are the steps for solving quadratic equations when squared terms are involved?
5. How would you graph $x(x^2 + x - 1)^2$ and $(4 + x^9)^2$?

Tip:

When simplifying complex fractions, always look for ways to factor or cancel out terms to make the process easier.