To solve the quadratic equation, we need to find the roots by factoring or using the quadratic formula.

Given equation: 4x^4 - 2x^3 - 14x^2 + 15x + 26 = 0

Let's factor the equation by grouping:

4x^4 - 2x^3 - 14x^2 + 15x + 26 = 0
4x^4 - 8x^3 + 6x^3 - 12x^2 - 2x^2 + 5x + 21x + 26 = 0
4x^3(x - 2) + 6x^2(x - 2) - 2(x - 2) + 5(x + 5) = 0
(4x^3 + 6x^2 - 2)(x - 2) + 5(x + 5) = 0
2x^2(2x+3)(x - 2) + 5(x + 5) = 0

Now we have factored the equation into (2x^2 + 3)(2x - 1)(x - 2) + 5(x + 5) = 0

Setting each factor to zero will give us the roots:

2x^2 + 3 = 0
2x^2 = -3
x^2 = -3/2
x = ±√(-3/2) -- imaginary roots

2x - 1 = 0
2x = 1
x = 1/2

x - 2 = 0
x = 2

x + 5 = 0
x = -5

Therefore, the roots of the equation are x = 1/2, x = 2, and x = -5.