The maximum height of a pole that an object following a path defined by y = -4x^2 + 6x + 2 can pass over is the maximum value of the function y = -4x^2 + 6x + 2.

This is obtained by differentiating y and solving y' = 0. Also, the value of y'' for the solution of y' = 0 should be negative.

y = -4x^2 + 6x + 2

y' = -8x + 6

y'' = -8 which is always negative

Solving y' = -8x + 6 = 0

=> 8x = 6

=> x = 6/8

At x = 3/4, y = `-4*(3/4)^2 + 6*(3/4) + 2`

=> `-4*(9/16) + 18/4 + 2`

=> `-9/4 + 18/4 + 2`

=> `9/4 + 2`

=> `17/4`

=> 4.25

**The maximum height of a pole that an object following a path defined by y = -4x^2 + 6x + 2 can pass over is 4.25 **

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