This is an equation involving rational expressions. We can simplify it step by step.

We are given:

$\frac{x^2 - 4}{x^2 + x - 6} = \frac{2x^2 - x - 10}{x^2 + 8x + 15} \cdot \frac{2x^2 + x - 15}{x^2 - 25}$

Step 1: Factor the quadratic expressions

• The numerator $x^2 - 4$ is a difference of squares, so it factors as: $x^2 - 4 = (x - 2)(x + 2)$

• The denominator $x^2 + x - 6$ factors as: $x^2 + x - 6 = (x - 2)(x + 3)$

• The numerator $2x^2 - x - 10$ factors as: $2x^2 - x - 10 = (2x + 5)(x - 2)$

• The denominator $x^2 + 8x + 15$ factors as: $x^2 + 8x + 15 = (x + 3)(x + 5)$

• The numerator $2x^2 + x - 15$ factors as: $2x^2 + x - 15 = (2x - 3)(x + 5)$

• The denominator $x^2 - 25$ is a difference of squares, so it factors as: $x^2 - 25 = (x - 5)(x + 5)$

Step 2: Substitute the factored terms into the equation

Now, substitute the factored expressions into the equation:

$\frac{(x - 2)(x + 2)}{(x - 2)(x + 3)} = \frac{(2x + 5)(x - 2)}{(x + 3)(x + 5)} \cdot \frac{(2x - 3)(x + 5)}{(x - 5)(x + 5)}$

Step 3: Cancel common factors

We can cancel the common factors in the numerator and denominator across both sides:

• On the left-hand side, $(x - 2)$ cancels from both the numerator and denominator.
• On the right-hand side, $(x + 5)$ cancels in two places, and $(x - 2)$ cancels from the numerator and denominator.

After canceling, we are left with:

$\frac{x + 2}{x + 3} = \frac{(2x + 5)(2x - 3)}{(x - 5)(x + 3)}$

Step 4: Cross multiply to simplify further

Now, cross-multiply to solve for $x$:

$(x + 2)(x - 5)(x + 3) = (2x + 5)(2x - 3)(x + 3)$

From here, we can expand both sides and solve for $x$. Would you like me to continue solving this equation or would you like further steps?