If 4x^3-12x^2-37x-15 = 0 has an integer root then it has a corresponding factor of the form (x+a) with a a divisor of the constant term -15. That gives possibilities of +-1, +-3, +-5 or +-15. If we try x=5 we find:
4x^3-12x^2-37x-15
= (4xx125)-(12xx25)-(37xx5)-15
= 500-300-185-15 = 0
So x=5 is a root of 4x^3-12x^2-37x-15 = 0 and (x-5) must be a factor of 4x^3-12x^2-37x-15.
Next use synthetic division to find:
4x^3-12x^2-37x-15 = (x-5)(4x^2+8x+3)
The remaining quadratic factor 4x^2+8x+3 is of the form ax^2+bx+c with a=4, b=8 and c=3.
This has discriminant
Delta = b^2 - 4ac = 8^2 - (4xx4xx3) = 64 - 48 = 16 = 4^2
...a nice positive square number, so 4x^2+8x+3 = 0 has 2 distinct rational roots, given by the formula
x = (-b+-sqrt(Delta))/(2a) = (-8+-4)/8 = (-2+-1)/2
Multiplying both sides by 2 we get
2x = -2+-1
Hence (2x+1) and (2x+3) are factors of 4x^2+8x+3
Putting this together:
4x^3-12x^2-37x-15 = (x-5)(2x+1)(2x+3)
