To solve the integral $\int_0^{\infty} 3 - 4z^2 \, dz$ using Beta and Gamma functions, let's first ensure that we understand the structure of the integral. The integral appears to involve the term $3 - 4z^2$, which doesn't directly fit the typical forms for Beta or Gamma functions. However, if this is indeed the integral, here’s an approach we might consider.

Given that: $\int_0^{\infty} 3 - 4z^2 \, dz$ doesn't naturally align with the form of Gamma or Beta functions, it’s helpful to clarify that the form might need to be adjusted or perhaps has a different intended integrand that resembles expressions used for Gamma or Beta functions.

Could you confirm if the expression is written as intended or if there's an adjustment that would make the use of Beta or Gamma functions clear?

If you need, I can guide you through understanding how to use Beta and Gamma functions with a different integral that matches their specific form, or if you have additional context.

Would you like a general explanation on using Beta and Gamma functions for integration? Here are related questions we can explore:

1. How can we rewrite integrals to match Beta and Gamma function forms?
2. What types of integrals are ideal for Gamma functions, specifically?
3. How are Beta and Gamma functions related?
4. How can substitutions simplify integrals for Beta function applications?
5. What are the steps to convert an integral into the form $\int_0^{\infty} x^{a-1} e^{-x} dx$?

Tip: Remember, Beta functions are often used in integrals over finite intervals, while Gamma functions typically apply to integrals from zero to infinity.