1. **State the problem:** Simplify the expression $$\frac{2x^4 + 2x^3 - 7x^2 + 5x + 10}{x^2 - 1}$$. 2. **Recall the formula and rules:** The denominator is a difference of squares: $$x^2 - 1 = (x - 1)(x + 1)$$. 3. **Factor the numerator if possible:** Try to factor or use polynomial division. 4. **Perform polynomial long division:** Divide numerator by denominator. Divide $$2x^4 + 2x^3 - 7x^2 + 5x + 10$$ by $$x^2 - 1$$. - First term: $$\frac{2x^4}{x^2} = 2x^2$$. - Multiply divisor by $$2x^2$$: $$2x^2(x^2 - 1) = 2x^4 - 2x^2$$. - Subtract: $$\left(2x^4 + 2x^3 - 7x^2 + 5x + 10\right) - \left(2x^4 - 2x^2\right) = 2x^3 - 5x^2 + 5x + 10$$. - Next term: $$\frac{2x^3}{x^2} = 2x$$. - Multiply divisor by $$2x$$: $$2x(x^2 - 1) = 2x^3 - 2x$$. - Subtract: $$\left(2x^3 - 5x^2 + 5x + 10\right) - \left(2x^3 - 2x\right) = -5x^2 + 7x + 10$$. - Next term: $$\frac{-5x^2}{x^2} = -5$$. - Multiply divisor by $$-5$$: $$-5(x^2 - 1) = -5x^2 + 5$$. - Subtract: $$\left(-5x^2 + 7x + 10\right) - \left(-5x^2 + 5\right) = 7x + 5$$. 5. **Write the result:** $$\frac{2x^4 + 2x^3 - 7x^2 + 5x + 10}{x^2 - 1} = 2x^2 + 2x - 5 + \frac{7x + 5}{x^2 - 1}$$ 6. **Final answer:** $$\boxed{2x^2 + 2x - 5 + \frac{7x + 5}{(x - 1)(x + 1)}}$$