1. **State the problem:** Simplify or analyze the quadratic expression $$x^2 - 4dx + 2d^2$$ where $x$ and $d$ are variables. 2. **Recall the quadratic form:** A quadratic expression in $x$ generally looks like $$ax^2 + bx + c$$ where $a$, $b$, and $c$ can be constants or expressions involving other variables. 3. **Identify coefficients:** Here, $$a = 1, \quad b = -4d, \quad c = 2d^2.$$ 4. **Check if it can be factored:** We look for two numbers that multiply to $$a \times c = 1 \times 2d^2 = 2d^2$$ and add to $$b = -4d$$. 5. The factors of $$2d^2$$ are $$-2d$$ and $$-2d$$, since $$-2d \times -2d = 4d^2$$ which is not equal to $$2d^2$$, so direct factoring is not straightforward. 6. **Use the quadratic formula to find roots:** $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-(-4d) \pm \sqrt{(-4d)^2 - 4 \times 1 \times 2d^2}}{2 \times 1} = \frac{4d \pm \sqrt{16d^2 - 8d^2}}{2} = \frac{4d \pm \sqrt{8d^2}}{2} $$ 7. Simplify the square root: $$ \sqrt{8d^2} = \sqrt{8} \times \sqrt{d^2} = 2\sqrt{2} |d| $$ 8. Substitute back: $$ x = \frac{4d \pm 2\sqrt{2} |d|}{2} = 2d \pm \sqrt{2} |d| $$ 9. **Final roots:** $$ x_1 = 2d + \sqrt{2} |d|, \quad x_2 = 2d - \sqrt{2} |d| $$ 10. **Summary:** The quadratic $$x^2 - 4dx + 2d^2$$ has roots $$2d \pm \sqrt{2} |d|$$ and cannot be factored easily with simple integers unless $d$ is specified. **Answer:** The roots of the quadratic are $$x = 2d \pm \sqrt{2} |d|$$.