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The problem asks to write a polynomial with the given zeros \(-3\), \(\sqrt{5}\), and integer coefficients, with a leading coefficient of 1.

### Step-by-Step Solution:

#### Step 1: Zeros of the Polynomial
The given zeros are:
- \( x = -3 \)
- \( x = \sqrt{5} \)

Since the polynomial must have integer coefficients, the conjugate of \(\sqrt{5}\), which is \(-\sqrt{5}\), must also be a root. So, the zeros are:
- \( x = -3 \)
- \( x = \sqrt{5} \)
- \( x = -\sqrt{5} \)

#### Step 2: Convert Zeros into Factors
For each zero, the corresponding factor of the polynomial is:
- For \( x = -3 \), the factor is \( (x + 3) \).
- For \( x = \sqrt{5} \), the factor is \( (x - \sqrt{5}) \).
- For \( x = -\sqrt{5} \), the factor is \( (x + \sqrt{5}) \).

#### Step 3: Multiply the Factors
Now, we'll multiply the factors together:

1. Multiply the conjugate pair \( (x - \sqrt{5})(x + \sqrt{5}) \). This is a difference of squares:
   \[
   (x - \sqrt{5})(x + \sqrt{5}) = x^2 - 5
   \]
   
2. Now, multiply this result by \( (x + 3) \):
   \[
   (x + 3)(x^2 - 5)
   \]

#### Step 4: Expand the Expression
Distribute \( (x + 3) \) across \( (x^2 - 5) \):
\[
(x + 3)(x^2 - 5) = x(x^2 - 5) + 3(x^2 - 5)
\]
Now, simplify:
\[
x(x^2 - 5) = x^3 - 5x
\]
\[
3(x^2 - 5) = 3x^2 - 15
\]

So, the expanded form is:
\[
x^3 - 5x + 3x^2 - 15
\]

#### Step 5: Write the Polynomial in Standard Form
Rearrange the terms in descending order of degree:
\[
x^3 + 3x^2 - 5x - 15
\]

Thus, the polynomial in standard form is:
\[
\boxed{x^3 + 3x^2 - 5x - 15}
\]

Would you like further details on any step? 

### Relative Questions:
1. How do you derive a polynomial when given complex roots?
2. Why do we include the conjugate of irrational roots?
3. What are the steps to factor a polynomial once it is expanded?
4. How does the leading coefficient affect the behavior of the polynomial?
5. Can you have a polynomial with only real zeros if irrational roots are included?

### Tip:
When given irrational or complex roots and asked for integer coefficients, always check if the conjugate of the root must be included for the polynomial to maintain integer values.

Write the polynomial with zeros of -3, sqrt(5), integer coefficients, and lead coefficient 1. Write your answer in standard form.

Learn how to write a polynomial in standard form with the given zeros of -3 and sqrt(5), including how to handle irrational roots and apply the difference of squares theorem for integer coefficients.

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