Read your first line, can you see how it can be misinterpreted?

so let

f(x) = 4x^4+8x^3-13x^2-2x+3

try x = ±1, ±3

f(1) = 0, so x-1 is a factor,

I then used synthetic division to show that

4x^4+8x^3-13x^2-2x+3 = (x-1)(4x^3 + 12x^2 - x - 3)

but (4x^3 + 12x^2 - x - 3)

= 4x^2(x+3) - (x+3)

= (x+3)(4x^1 - 1)

= (x+3)*2x-1)(2x+1)

so for

4x^4+8x^3-13x^2-2x+3 = 0

(x-1)x+3)(2x+1)(2x-1) = 0

x = 1, -3, ±1/2