Find zeroes of the given quadratic polynomials and verify the relation between zeroes and coefficients :
`sqrt(3)x^(2)+8x+5sqrt(3)`

Explanation

To find the zeroes of the quadratic polynomial \( \sqrt{3}x^2 + 8x + 5\sqrt{3} \) and verify the relation between the zeroes and coefficients, we will follow these steps:

### Step 1: Identify coefficients
The given polynomial can be expressed in the standard form \( ax^2 + bx + c \), where:
- \( a = \sqrt{3} \)
- \( b = 8 \)
- \( c = 5\sqrt{3} \)

### Step 2: Calculate the discriminant
The discriminant \( D \) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = 8^2 - 4 \cdot \sqrt{3} \cdot 5\sqrt{3} \]
Calculating this:
\[ D = 64 - 4 \cdot 3 \cdot 5 = 64 - 60 = 4 \]

### Step 3: Find the zeroes using the quadratic formula
The zeroes (roots) of the polynomial can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values of \( b \), \( D \), and \( a \):
\[ x = \frac{-8 \pm \sqrt{4}}{2\sqrt{3}} \]
Calculating further:
\[ x = \frac{-8 \pm 2}{2\sqrt{3}} \]

### Step 4: Calculate the two roots
1. For the first root \( x_1 \):
\[ x_1 = \frac{-8 + 2}{2\sqrt{3}} = \frac{-6}{2\sqrt{3}} = \frac{-3}{\sqrt{3}} = -\sqrt{3} \]

2. For the second root \( x_2 \):
\[ x_2 = \frac{-8 - 2}{2\sqrt{3}} = \frac{-10}{2\sqrt{3}} = \frac{-5}{\sqrt{3}} \]

### Step 5: Verify the relation between zeroes and coefficients
The sum of the roots \( (x_1 + x_2) \) and the product of the roots \( (x_1 \cdot x_2) \) can be verified against the coefficients:
- The sum of the roots \( x_1 + x_2 \):
\[ x_1 + x_2 = -\sqrt{3} + \left(-\frac{5}{\sqrt{3}}\right) = \frac{-3 - 5}{\sqrt{3}} = \frac{-8}{\sqrt{3}} \]
According to Vieta's formulas, this should equal \( -\frac{b}{a} \):
\[ -\frac{b}{a} = -\frac{8}{\sqrt{3}} \]

- The product of the roots \( x_1 \cdot x_2 \):
\[ x_1 \cdot x_2 = \left(-\sqrt{3}\right) \left(-\frac{5}{\sqrt{3}}\right) = \frac{5}{3} \]
According to Vieta's formulas, this should equal \( \frac{c}{a} \):
\[ \frac{c}{a} = \frac{5\sqrt{3}}{\sqrt{3}} = 5 \]

### Conclusion
Thus, we have found the zeroes of the polynomial:
- \( x_1 = -\sqrt{3} \)
- \( x_2 = -\frac{5}{\sqrt{3}} \)

And we verified the relations:
- Sum of roots \( = -\frac{b}{a} \)
- Product of roots \( = \frac{c}{a} \)