Question

what is the root of 4x to the 4th power + 8x to the 3rd power - 13x to the 2nd power -2x +3

Answer 1

I think you mean
(√(4x))^4 + 8x^3 - 13x^2 - 2x + 3
which is
16x^2 + 8x^3 - 13x^2 - 2x + 3
= 8x^3 + 3x^2 - 2x + 3

Now what about it?
What do want done with it?

Answer 2

I need the roots of 4x^4+8x^3-13x^2-2x+3

Answer 3

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