Parallel lines have the same slope. Once you know the slope of the original line, your new line must use that exact slope.

Plug the slope and the point $(3,-2)$ into point-slope form $y - y_1 = m(x - x_1)$, then rearrange the equation into $y=mx+b$ so you can compare it with the answer choices.

In a new expression line, type y=2x-3 to graph a line equivalent to $4x-2y=6$. This helps you see the slope and orientation of the given line.

In separate expression lines, enter each choice exactly as written: y=-1/2x+1/2, y=2x+8, y=-2x-8, and y=2x-8. You should now see four candidate lines along with the original line.

Add the point (3,-2) in Desmos. Look for which of the four candidate lines passes through this point and has the same tilt (slope) as the original line y=2x-3. The equation of that line is the correct choice.

Rewrite the given equation $4x-2y=6$ in slope-intercept form $y=mx+b$.

Start by isolating $y$:

So the slope of the given line is $m=2$.

Any line parallel to the given line must have the same slope $m=2$.

Use point-slope form with the point $(3,-2)$ and slope $2$:

So we have:

Now simplify $y + 2 = 2(x - 3)$ to get it into $y=mx+b$ form:

This matches answer choice D) $y=2x-8$, so that is the equation of the line that is parallel to $4x-2y=6$ and passes through $(3,-2)$.