1. **State the problem:** Simplify or analyze the quadratic expression $2x^2 - 8x - 7$. 2. **Formula and rules:** A quadratic expression is generally written as $ax^2 + bx + c$. 3. **Identify coefficients:** Here, $a=2$, $b=-8$, and $c=-7$. 4. **Factorization:** To factor, find two numbers that multiply to $a \times c = 2 \times (-7) = -14$ and add to $b = -8$. 5. The numbers are $-7$ and $2$ because $-7 \times 2 = -14$ and $-7 + 2 = -5$ (not $-8$), so direct factoring is not straightforward. 6. Use the quadratic formula to find roots: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \times 2 \times (-7)}}{2 \times 2}$$ 7. Calculate discriminant: $$64 + 56 = 120$$ 8. Roots: $$x = \frac{8 \pm \sqrt{120}}{4} = \frac{8 \pm 2\sqrt{30}}{4} = 2 \pm \frac{\sqrt{30}}{2}$$ 9. So the factorization is: $$2x^2 - 8x - 7 = 2\left(x - \left(2 + \frac{\sqrt{30}}{2}\right)\right)\left(x - \left(2 - \frac{\sqrt{30}}{2}\right)\right)$$ 10. This shows the roots and factorization of the quadratic expression.