1/(x^2 + 3x) + 1/(x^2 + x) / (x/(x^2 - 1) - x/(x + 3))

Steps to Solve

1. Factor the denominators

We start by factoring the denominators of the fractions:

• $x^2 + 3x = x(x + 3)$
• $x^2 + x = x(x + 1)$
• $x^2 - 1 = (x - 1)(x + 1)$ (difference of squares)
• The denominators are $x(x + 3)$, $x(x + 1)$, $(x - 1)(x + 1)$, and $(x + 3)$.

2. Finding the common denominator

The least common denominator (LCD) for all fractions appears to be: $$ \text{LCD} = x(x + 3)(x + 1)(x - 1) $$

3. Rewrite each fraction with the common denominator

Now we rewrite each of the fractions so they have the same denominator:

For the first component: $$ \frac{1}{x(x + 3)} = \frac{1 \cdot (x + 1)(x - 1)}{x(x + 3)(x + 1)(x - 1)} $$

For the second component: $$ \frac{1}{x(x + 1)} = \frac{1 \cdot (x + 3)(x - 1)}{x(x + 1)(x + 3)(x - 1)} $$

For the numerator of the second part: $$ \frac{x}{(x - 1)(x + 1)} = \frac{x \cdot (x + 3)}{(x - 1)(x + 1)(x + 3)} $$

And for the denominator of the second part: $$ \frac{x}{(x + 3)} = \frac{x \cdot (x - 1)(x + 1)}{(x + 3)(x - 1)(x + 1)} $$

4. Combine the fractions on the numerator

Now, we combine the fractions in the numerator: $$ \frac{(x + 1)(x - 1) + (x + 3)(x - 1)}{LCD} $$

5. Simplify the expression

Combine like terms in the numerator: $$ = \frac{(x^2 - 1) + (x^2 - x + 3x - 3)}{LCD} $$ $$ = \frac{2x^2 + 2}{LCD} $$

6. Combine the fractions in the second part

Now, we simplify and combine the two fractions from the second part, which are: $$ \frac{x(x + 3)}{(x - 1)(x + 1)(x + 3)} - \frac{x(x - 1)(x + 1)}{(x + 3)(x - 1)(x + 1)} $$

7. Combine the two fractions

Now, combine these as: $$ \frac{x(x + 3) - x(x - 1)(x + 1)}{(x - 1)(x + 1)(x + 3)} $$ After expanding and simplifying, we calculate the final form.

8. Final subtraction

Subtract the combined numerator results from step 5 and use the LCD: $$ \frac{2x^2 + 2 - (x^2 + 2)}{LCD} $$

And finally simplify the entire complex fraction.