obtain all the zeroes of `3x^4+6x^3-2x^2-10x-5` if two of its zeroes are `sqrt(5/3)` and `-sqrt(5/3)`

Explanation

To find all the zeroes of the polynomial \( p(x) = 3x^4 + 6x^3 - 2x^2 - 10x - 5 \) given that two of its zeroes are \( \sqrt{\frac{5}{3}} \) and \( -\sqrt{\frac{5}{3}} \), we can follow these steps:

### Step 1: Write the Known Roots as Factors
Since we know two roots, we can express them as factors of the polynomial:
\[ (x - \sqrt{\frac{5}{3}})(x + \sqrt{\frac{5}{3}}) = x^2 - \left(\sqrt{\frac{5}{3}}\right)^2 = x^2 - \frac{5}{3} \]

### Step 2: Multiply by 3 to Eliminate the Fraction
To make calculations easier, we can multiply the factor by 3:
\[ 3\left(x^2 - \frac{5}{3}\right) = 3x^2 - 5 \]

### Step 3: Perform Polynomial Long Division
Now, we will divide the polynomial \( p(x) \) by \( 3x^2 - 5 \) to find the other factor \( q(x) \).

1. Divide \( 3x^4 \) by \( 3x^2 \) to get \( x^2 \).
2. Multiply \( 3x^2 - 5 \) by \( x^2 \):
\[ 3x^4 - 5x^2 \]
3. Subtract this from \( p(x) \):
\[ (3x^4 + 6x^3 - 2x^2 - 10x - 5) - (3x^4 - 5x^2) = 6x^3 + 3x^2 - 10x - 5 \]

4. Now divide \( 6x^3 \) by \( 3x^2 \) to get \( 2x \).
5. Multiply \( 3x^2 - 5 \) by \( 2x \):
\[ 6x^3 - 10x \]
6. Subtract this from the current polynomial:
\[ (6x^3 + 3x^2 - 10x - 5) - (6x^3 - 10x) = 3x^2 - 5 \]

7. Now, divide \( 3x^2 \) by \( 3x^2 \) to get \( 1 \).
8. Multiply \( 3x^2 - 5 \) by \( 1 \):
\[ 3x^2 - 5 \]
9. Subtract this from the current polynomial:
\[ (3x^2 - 5) - (3x^2 - 5) = 0 \]

### Step 4: Write the Result of the Division
The result of the division gives us:
\[ p(x) = (3x^2 - 5)(x^2 + 2x + 1) \]

### Step 5: Factor the Quadratic
Now we need to factor \( x^2 + 2x + 1 \):
\[ x^2 + 2x + 1 = (x + 1)^2 \]

### Step 6: Find All Zeroes
Now we can find all the zeroes of \( p(x) \):
1. From \( 3x^2 - 5 = 0 \):
\[ 3x^2 = 5 \implies x^2 = \frac{5}{3} \implies x = \pm \sqrt{\frac{5}{3}} \]

2. From \( (x + 1)^2 = 0 \):
\[ x + 1 = 0 \implies x = -1 \quad (\text{with multiplicity 2}) \]

### Final Answer
Thus, the zeroes of the polynomial \( p(x) \) are:
\[ \sqrt{\frac{5}{3}}, -\sqrt{\frac{5}{3}}, -1, -1 \]