x3/ x3+27+9x(x+3) = x-3 / x+6

We are given that x³/(x+3)³ = x-3/(x+6). From componendo-dividendo, we get

(x+3)³-x³/(x+3)³+x³ = (x+6) - (x-3)/(x+6) + (x-3)

or 3 [x²+(x+3)² + x(x+3)]/(2x+3)(x²+(x+3)²-x(x+3)] = 9/(2x+3)

So, either x = -3/2 or 3[x²+(x+3)²-x(x+3)] = x²+(x+3)² + x(x+3)

The second equation would imply 9 = 0 and hence x = -3/2 is the only possibility