For what values of (m) does the graph of y=mx²-5x-2 have no x-intercepts?

You need to find where the discriminant is less than zero

Remember the discriminant is b^2 - 4ac

a = m

b = - 5

c = - 2

Set up the inequality like this

(-5) ^2 - 4m (-2) < 0

25 + 8m < 0

8m < - 25

m < - 25/8 < - 3.125

So any number less than - 3.125 will make this graph have no x-intercepts (or real roots)