want to make sure that I am using my vocabulary words correctly to explain my answer for this problem. Thank You!!!

). This Quadratic formula needs to be factored -x^2 + 3 x + 5 = 0

x^2 - 3 x - 5 = 0

x = [ +3 +/- sqrt(9 + 20) ] / 2 The discriminant

1.5 +/- 2.69

You can NOT factor that polynomial.

You must either complete the square or equivalently use the quadratic equation
you used the quadratic equation
which is
x = [ -b+/-sqrt(discriminant) ]/2a

the discriminant is
b^2 - 4 ac
which is
9 + 20 = 29