To solve this equation, we first simplify the right side of the equation:

(-1/2)^2 - 1
= (1/4) - 1
= 1/4 - 4/4
= -3/4

Now, we substitute this value back into the equation:

3x/4 + 5x/6 = -3/4

To solve for x, we need to find a common denominator for the fractions on the left side of the equation. The common denominator for 4 and 6 is 12.

Multiplying the first fraction by 3/3 and the second fraction by 2/2 to get a common denominator of 12:

9x/12 + 10x/12 = -3/4

Combining the fractions:

(9x + 10x) / 12 = -3/4
19x / 12 = -3/4

Now, we can solve for x by multiplying both sides by 12:

19x = -3 * 12
19x = -36
x = -36 / 19

Therefore, the solution to the equation is x = -36/19.