Obtain all other zeroes of 3x⁴ + 6x³ - 2x² - 10x - 5, if two of its zeroes are √5/3 and -√5/3.

Two zeroes are √5/3 and -√5/3
So we can write it as,
x = √5/3 and x = -√5/3
we get
x - √5/3 = 0 and x + √5/3 = 0
Multiply both the factors we get,
x² - 5/3 = 0
Multiply by 3
we get 3x² - 5 = 0
is the factor of 3x⁴ + 6x³ - 2x² - 10x - 5
Now divide 3x⁴ + 6x³ - 2x² - 10x - 5 by 3x² - 5 = 0
we get,
Quotient is x² + 2x + 1 = 0
Compare the equation with quadratic formula,
x² - (Sum of root)x + (Product of root) = 0
⇒Sum of root = -2
⇒Product of the root = 1
So, we get
⇒x² + x + x + 1 = 0
⇒x(x + 1) + 1(x + 1) = 0
⇒(x + 1)(x + 1) = 0
⇒x + 1 = 0, x + 1 = 0
⇒x = -1, x = -1
So, our zeroes are -1, -1, √5/3 and -√5/3