y = ax^4 + bx^3 + cx^2 + dx + e.

Substituting the given values:

a(-2)^4 + b(-2)^3 + c(-2)^2 + d(-2) + e = 45

16a – 8b + 4c – 2d + e = 45 …(1)

a(-1)^4 + b(-1)^3 + c(-1)^2 + d(-1) + e = 48

a – b + c – d + e = 48 …(2)

a0^4 + b0^3 + c0^2 + d(0) + e = 45

e = 45 …(3)

a(1)^4 + b(1)^3 + c(1)^2 + d(1) + e = 48

a + b + c + d + e = 48 …(4)

a2^4 + b2^3 + c2^2 + d2 + e = 45

16a + 8b + 4c + 2d + e = 45 …(5)

From (3):

e = 45

The equations can therefore be reduced to:

16a – 8b + 4c – 2d = 0 …(6)

a – b + c – d = 3 …(7)

a + b + c + d = 3 …(8)

16a + 8b + 4c + 2d = 0 …(9)

Adding (7) & (8):

2a + 2c = 6

a + c = 3 …(10)

Adding (6) & (9):

32a + 8c = 0

4a + c = 0 …(11)

Subtracting (11) from (10):

-3a = 3

a = -1

From (11):

c = 4

Substituting for a & c in (7) & (9):

– b – d = 0

4b + d = 0

Hence b = 0, d = 0

The equation is therefore:

y = -x^4 + 4x^2 + 45.