To find the quadratic polynomial with given zeroes, we follow a systematic approach. Here’s a detailed explanation.

Understanding Zeroes of a Polynomial
- Zeroes are the values of \( x \) for which the polynomial equals zero.
- For a quadratic polynomial of the form \( ax^2 + bx + c \), the zeroes can be represented as \( p \) and \( q \).

Using the Given Zeroes
- In this case, the zeroes are \( -3 \) and \( -5 \).
- The quadratic polynomial can be constructed using the formula:
\[
P(x) = a(x - p)(x - q)
\]
- Here, \( p = -3 \) and \( q = -5 \).

Substituting the Values
- Substitute the zeroes into the polynomial formula:
\[
P(x) = a(x + 3)(x + 5)
\]
- To simplify, we can expand the expression:
\[
P(x) = a[(x + 3)(x + 5)]
\]

Expanding the Product
- Using the distributive property (FOIL), we expand:
\[
= a[x^2 + 5x + 3x + 15]
\]
- This simplifies to:
\[
= a[x^2 + 8x + 15]
\]

Choosing the Leading Coefficient
- Typically, we choose \( a = 1 \) for simplicity:
\[
P(x) = x^2 + 8x + 15
\]

Final Polynomial
- The quadratic polynomial whose zeroes are \( -3 \) and \( -5 \) is:
\[
\boxed{x^2 + 8x + 15}
\]