Question 323495
Hi Tom,
You were correct in using the FOIL method for this problem, you will use this method 3 times.  
First rewrite the problem so it is easier to see how to multiply:
(x-2)^2(x+1)(x-5)
=(x-2)(x-2)(x+1)(x-5)
Now we will use the method 3 times.  First, multiply out just the (x-2)(x-2):
(x-2)(x-2)(x+1)(x-5)
=(x^2-2x-2x+4)(x+1)(x-5)
=(x^2-4x+4)(x+1)(x-5)
Now multiply (x^2-4x+4)(x+1) using FOIL:
=(x^2-4x+4)(x+1)(x-5)
=(x^3-4x^2+4x+x^2-4x+4)(x-5)
This can be simplified by adding the x^2 values and adding the x values:
=(x^3-3x^2+4)(x-5)
Finally, multiply (FOIL) what is left:
=(x^3-3x^2+4)(x-5)
=(x^4-3x^3+4x-5x^3+15x^2-20)
=(x^4-8x^3+15x^2+4x-20)
So this is your simplified polynomial form:
x^4-8x^3+15x^2+4x-20= 0

I hope this helped!  Let me know if you have more questions!