To solve the rational equation 2/(x + 2) = 9/(8 - 5x)/(4x + 8), we first simplify the right side by multiplying the numerator and denominator by the LCD of 4x + 8:
2/(x + 2) = 9(4x + 8)/(8 - 5x)
2/(x + 2) = (36x + 72)/(5x - 8)
Now we can cross-multiply and simplify:
2(5x - 8) = (x + 2)(36x + 72)
10x - 16 = 36x^2 + 80x + 144
36x^2 + 70x + 160 = 0
We can simplify this quadratic equation by dividing both sides by 2:
18x^2 + 35x + 80 = 0
Now we can use the quadratic formula to solve for x:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
where a = 18, b = 35, and c = 80:
x = (-35 ± sqrt(35^2 - 4(18)(80))) / 2(18)
x = (-35 ± sqrt(137)) / 36
Therefore, the solutions to the rational equation 2/(x + 2) = 9/(8 - 5x)/(4x + 8) are:
x = (-35 + sqrt(137)) / 36
x = (-35 - sqrt(137)) / 36