The answer key is wrong.

To check the solution plug the value for x = 17/19 into the equation: 

-4(3x-5)=7x+3

-4(3*(17/19) - 5) = 7*(17/19) + 3

-12*(17/19) + 20 = 7*(17/19) + 3

(-12*17 + 19*20)/19 = (7*17 + 19*3)/19

176/19 = 176/19

The left side of the equation is equal to the right side of the equation follow that x = 17/19 is indeed the solution for the equation -4(3x-5)=7x+3.

Now we will prove the x = 3/5 is not a solution for -4(3x-5)=7x+3.

Plug x = 3 into -4(3x-5)=7x+3:

-4(3*(3/5)-5)=7*(3/5)+3

-4(9/5 - 5) = 21/5 + 3

-36/5 + 20 = (21 + 3*5)/5

(-36 + 20*5)/5 = 36/5

64/5 = 36/5 <- this statement is false (left side is different than the right side) thus x = 3/5 is not a solution for the equation -4(3x-5)=7x+3.