Determine the coordinates of the points where the curve $x^2-2xy+4y^2=8$ intersects the $x$-axis, and find the slopes of the curve at these points.

Points: $(2\(\sqrt{2}\), 0)$ and $(-2\(\sqrt{2}\), 0)$

Slope at both points: $1$

Points: $(2\(\sqrt{2}\), 0)$ and $(-2\(\sqrt{2}\), 0)$

Slope at both points: $-1$

Points: $(\sqrt{2},0)$ and $(-2\(\sqrt{2}\), 0)$

Slope at $(\sqrt{2},0)$: $-1$

Slope at $(-2\(\sqrt{2}\), 0)$: $1$

Points: $(\(\sqrt{2}\),0)$ and $(-2\(\sqrt{2}\), 0)$

Slope at $(\(\sqrt{2}\),0)$: $1$

Slope at $(-2\(\sqrt{2}\), 0)$: $-1$