4. (x + 2)(x - 2) / (x - 1)(x + 1) = 4 / 5

Steps to Solve

1. Identify the given equation The equation is given as: $$ \frac{(x+2)(x-2)}{x^2} = \frac{(x-1)(x+1)}{5} $$

2. Cross-multiply to eliminate fractions To eliminate the fractions, we can cross-multiply: $$ 5(x+2)(x-2) = (x-1)(x+1)x^2 $$

3. Expand both sides Now we need to expand both sides of the equation: Left side: $$ 5((x+2)(x-2)) = 5(x^2 - 4) = 5x^2 - 20 $$

Right side: $$ (x-1)(x+1) = x^2 - 1 $$ Thus, $$ (x^2 - 1)x^2 = x^4 - x^2 $$

4. Set the equation to zero Combine the terms: $$ 5x^2 - 20 = x^4 - x^2 $$

Rearranging gives: $$ 0 = x^4 - 6x^2 + 20 $$

5. Form a standard polynomial Rearranging the equation gives: $$ x^4 - 6x^2 + 20 = 0 $$

6. Use substitution for quadratic form Let $y = x^2$. The equation becomes: $$ y^2 - 6y + 20 = 0 $$

7. Apply the quadratic formula Using the quadratic formula (y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}): For the current equation, where (a = 1), (b = -6), and (c = 20): $$ y = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(20)}}{2(1)} $$

8. Calculate the discriminant Calculate the discriminant: $$ (-6)^2 - 4(1)(20) = 36 - 80 = -44 $$

Since the discriminant is negative, there are no real solutions for (y), and consequently no real solutions for (x).