\( 1 \leftarrow \left\lvert\, \begin{array}{l}\begin{array}{l}\text { Add. } \\ \frac{4}{5} x^{2}+\frac{1}{3} x+\frac{1}{4} \\ \frac{\frac{4}{5} x^{2}-\frac{2}{3} x+\frac{1}{6}}{} \\ \text { The sum is } \square . \\ \text { (Simplify your answer. Type coefficients as integers or fractions.) }\end{array}\end{array}\right. \)

To add the given polynomials, we must first align the terms properly. The two expressions provided are: 1. \(\frac{4}{5} x^{2} + \frac{1}{3} x + \frac{1}{4}\) 2. \(\frac{4}{5} x^{2} - \frac{2}{3} x + \frac{1}{6}\) Now, we will combine like terms: - For the \(x^2\) terms: \(\frac{4}{5} + \frac{4}{5} = \frac{8}{5} x^{2}\) - For the \(x\) terms: \(\frac{1}{3} - \frac{2}{3} = -\frac{1}{3} x\) - For the constant terms: \(\frac{1}{4} + \frac{1}{6}\). To add these fractions, we need a common denominator, which is 12. Thus: - \(\frac{1}{4} = \frac{3}{12}\) - \(\frac{1}{6} = \frac{2}{12}\) - Therefore, \(\frac{3}{12} + \frac{2}{12} = \frac{5}{12}\) Putting all this together, the sum is: \[ \frac{8}{5} x^{2} - \frac{1}{3} x + \frac{5}{12} \] This is the simplified form of the sum!