Combine: \( \left(\frac{1}{3} x^{2}-\frac{5}{6} x+\frac{3}{4}\right)-\left(\frac{7}{12} x^{2}+\frac{1}{2} x-\frac{2}{3}\right) \)

To combine the expressions, we start by distributing the negative sign through the second parentheses: \[ \left(\frac{1}{3} x^{2} - \frac{5}{6} x + \frac{3}{4}\right) - \left(\frac{7}{12} x^{2} + \frac{1}{2} x - \frac{2}{3}\right) = \frac{1}{3} x^{2} - \frac{5}{6} x + \frac{3}{4} - \frac{7}{12} x^{2} - \frac{1}{2} x + \frac{2}{3} \] Now, we will combine like terms. First, let’s work with the \(x^2\) terms: \[ \frac{1}{3} x^{2} - \frac{7}{12} x^{2} \] To combine these, we need a common denominator. The least common multiple of 3 and 12 is 12. Thus: \[ \frac{1}{3} = \frac{4}{12} \] So, \[ \frac{4}{12} x^{2} - \frac{7}{12} x^{2} = \frac{(4 - 7)}{12} x^{2} = -\frac{3}{12} x^{2} = -\frac{1}{4} x^{2} \] Now for the \(x\) terms: \[ -\frac{5}{6} x - \frac{1}{2} x = -\frac{5}{6} x - \frac{3}{6} x = -\frac{8}{6} x = -\frac{4}{3} x \] Finally, we combine the constant terms: \[ \frac{3}{4} + \frac{2}{3} \] To combine these, we need to find the least common multiple of 4 and 3, which is 12. Thus: \[ \frac{3}{4} = \frac{9}{12}, \quad \frac{2}{3} = \frac{8}{12} \] Combining these gives: \[ \frac{9}{12} + \frac{8}{12} = \frac{17}{12} \] Putting it all together, we have: \[ -\frac{1}{4} x^{2} - \frac{4}{3} x + \frac{17}{12} \] So the final combined expression is: \[ -\frac{1}{4} x^{2} - \frac{4}{3} x + \frac{17}{12} \]