\( (8 x-9)^{2}+3(8 x-9)-4=0 \) The solution set is \( \{\square\} \).

To solve the equation \( (8x-9)^{2}+3(8x-9)-4=0 \), let's introduce a substitution: let \( y = 8x - 9 \). Then the equation becomes \( y^2 + 3y - 4 = 0 \). To solve for \( y \), we can use the quadratic formula, \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1, b = 3, c = -4 \). Calculating the discriminant: \[ b^2 - 4ac = 3^2 - 4(1)(-4) = 9 + 16 = 25. \] Now, substituting back into the formula gives us: \[ y = \frac{-3 \pm \sqrt{25}}{2(1)} = \frac{-3 \pm 5}{2}. \] This leads to two solutions: 1. \( y = \frac{2}{2} = 1 \) 2. \( y = \frac{-8}{2} = -4 \) Now we revert back to \( x \): 1. For \( y = 1 \): \( 8x - 9 = 1 \) → \( 8x = 10 \) → \( x = \frac{10}{8} = \frac{5}{4} \). 2. For \( y = -4 \): \( 8x - 9 = -4 \) → \( 8x = 5 \) → \( x = \frac{5}{8} \). Thus, the solution set is \( \left\{ \frac{5}{4}, \frac{5}{8} \right\} \).