To find the roots of the given equation, we need to set each factor equal to zero and solve for x.

Setting [tex]x + 1 = 0 [/tex], we get [tex]x = -1[/tex].

Now, setting [tex]6x^{2} - 5x + 1 = 0[/tex], we can factor this quadratic equation as:

[tex]6x^{2} - 5x + 1 = (3x - 1)(2x - 1) = 0[/tex]

Setting each factor equal to zero, we have:

[tex]3x - 1 = 0 \implies 3x = 1 \implies x = \frac{1}{3}[/tex]

and

[tex]2x - 1 = 0 \implies 2x = 1 \implies x = \frac{1}{2}[/tex]

Therefore, the roots of the equation are [tex]x = -1, \frac{1}{3}, \frac{1}{2}[/tex].