To solve this quadratic by completing the square, you first need to get your quadratic to the form
color(blue)(x^2 + b/a = -c/a)
In your case, this means adding 2 to both sides of the equation and dividing everything by 5
5x^2 + 8x - color(red)cancelcolor(black)(2) + color(red)cancelcolor(black)(2) = 0 + 2
(color(red)cancelcolor(black)(5)x^2)/color(red)cancelcolor(black)(5) + 8/5x = 2/5
x^2 + 8/5x = 2/5
Next, use the coefficient of the x-term to help you determine what term needs to be added to both sides of the equation so that the left side can be written as the square of a binomial.
More specifically, divide said coefficient by 2, square the result, then add it to both sides of the equation
(8/5 * 1/2)^2 = 16/25
Your equation will now look like this
x^2 + 8/5x + 16/25 = 2/5 + 16/25
The left side of the equation can be written as
x^2 + 2 * (4/5) * x + (4/5)^2 = (x + 4/5)^2
This means that you now have
(x + 4/5)^2 = 26/25
Take the square root of both sides to get
sqrt( (x+4/5)^2) = sqrt(26/25)
x + 4/5 = +- sqrt(26)/5 implies x_(1,2) = -4/5 +- sqrt(26)/5
The two solutions to your quadratic will be
x_1 = color(green)(-4/5 - sqrt(26)/5) and x_2 = color(green)(-4/5 + sqrt(26)/5)