1. **State the problem:** Solve for $t$ in the equation $$-12 \sqrt{-57t + 64} + 64 + 21 = -75.$$\n\n2. **Simplify the constants on the left side:** $$64 + 21 = 85,$$ so the equation becomes $$-12 \sqrt{-57t + 64} + 85 = -75.$$\n\n3. **Isolate the square root term:** Subtract 85 from both sides:\n$$-12 \sqrt{-57t + 64} + 85 - 85 = -75 - 85,$$\nwhich simplifies to $$-12 \sqrt{-57t + 64} = -160.$$\n\n4. **Divide both sides by -12 to solve for the square root:**\n$$\cancel{-12} \sqrt{-57t + 64} = \frac{-160}{\cancel{-12}},$$\nwhich simplifies to $$\sqrt{-57t + 64} = \frac{160}{12} = \frac{40}{3}.$$\n\n5. **Square both sides to eliminate the square root:**\n$$\left(\sqrt{-57t + 64}\right)^2 = \left(\frac{40}{3}\right)^2,$$\nwhich gives $$-57t + 64 = \frac{1600}{9}.$$\n\n6. **Isolate the term with $t$:** Subtract 64 from both sides:\n$$-57t + 64 - 64 = \frac{1600}{9} - 64,$$\nwhich simplifies to $$-57t = \frac{1600}{9} - \frac{576}{9} = \frac{1024}{9}.$$\n\n7. **Divide both sides by -57 to solve for $t$:**\n$$t = \frac{\frac{1024}{9}}{-57} = -\frac{1024}{9 \times 57} = -\frac{1024}{513}.$$\n\n8. **Final answer:** $$t = -\frac{1024}{513}.$$