Please try leaving out the "I'm so bad at math" part. It doesn't help.
The given inequality is 4x + y > −6. Because of the > symbol, we must use a dashed line, not a solid one. Let's solve this for y: y > -4x - 6.
This (dashed) line has a slope of -4 and a y-intercept of (0,-6).
Plot (0,-6). Then, with your pencil pt. on that, move 1 unit to the right and then 4 units down. Plot a new point there, at (1, -10). Draw your dashed line thru those 2 pts. Then, because we have y > -4x - 6, shade the area ABOVE this line. Done.