What is the slope of a line parallel to this line 4x+5y=-3

First put it into slope intercept form. which is y=mx+b, in other words solve for y and get y by itself.
4x + 5y = -3
5y = -4x -3
y = -4/5x -3/5

The slope of any line in slope intercept form is the m, or the coefficient to x.
In this line the slope will be -4/5.

Now we find the slope of a line parallel to this line. Parallel lines have the same slope,
So the slope of a line parallel to 4x +5y = -3 is -4/5

4x + 5y = -3

5y = -4x - 3

y =  (-4/5)x - 3/5

Slopes of parallel lines are always the same.

The slope will be -4/5.