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Hello and welcome to JustAnswer,

The points of intersection can be found for those values x when:

1/2 x³ + x² + 1 = 9x² - 57/2 x +1

=> 1/2 x³ - 8x² + 57/2x = 0

=> x (x² - 4x + 57) = 0

=> x = 0 (this is the only solution, since x²-4x+57 is greater than 0 for every x)

So there is 1 point of intersection for these 2 curves: the point (0,1)

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Best regards,

Tom