1.Determine if x^2+5=0 mod 293 is solvable

I have x^2=-5 mod 293

So I look at (-5/293)=(-1/293)(5/293)=1*(5/293)=(293/5)=(3/5)=(5/3)=(2/3)=-1

So not solvable.

2. Determine if x^2+3x-7=0 mod 53 is solvable

I multiple by 4 first

4x^2+12x-28=0 mod 53

Then I have (2x+3)^2-19=0 mod 53

(2x+3)^2=19 mod 53

I look at (19/53)=(53/19)=(15/19)=(5/19)(3/19)=(5/19)(-1)=-(19/5)=-(4/5)=-(2/5)(2/5)=-(-1)(-1)=-1

So not solvable

Is this the right way of doing things?