help me please? i desperately need help with my math?

I assume you mean e^(- x^2). Did you want the first 4 terms including or excluding zeroes?

The Maclaurin series expansion for f(x) is

f(x) = f(0) + x f ' (0) + x^2/2! f ' ' (0) + ...

If f(x) = e^(- x^2) then

f ' (x) = - 2x e^(- x^2)

and f ' (0) = 0

f ' ' (x) = (4x^2 - 2) e^(- x^2)

and f ' ' (0) = - 2

f ' ' ' (x) = (- 8x^3 + 12x) e^(- x^2)

and f ' ' ' (0) = 0

So the Maclaurin series expansion for e^(- x^2) is

e^(- x^2) = 1 + 0 - x^2 + 0 ...

For further terms, just continue the process above.

P.S. It has just occurred to me that there is a simpler way.

Let - x^2 = y and do the expansion for g(y) = e^y as follows:

g(y) = g(0) + y g ' (0) + y^2/2! g ' ' (0) + ...

= 1 + y + y^2/2! + y^3/3! + ...

Then substitute - x^2 for y as follows:

e^(- x^2) = 1 - x^2 + x^4/2 - x^6/6 + ...