The expression given is:

\[
\ln\left(\left|6 + t + t^3\right|\right)
\]

Here's a breakdown of this expression:

1. **Inside the Absolute Value**: The expression \( 6 + t + t^3 \) could potentially be positive or negative depending on the value of \( t \). That's why the absolute value function \( |\cdot| \) is applied.

2. **Logarithm of the Absolute Value**: The natural logarithm function \( \ln \) is defined only for positive arguments. Thus, the absolute value guarantees that \( 6 + t + t^3 \) becomes positive, allowing the \( \ln \) function to be applied.

3. **Domain Analysis**: To determine the domain of the function, we must find where \( |6 + t + t^3| \) is greater than zero. This occurs whenever \( 6 + t + t^3 \neq 0 \). 

### Simplified Domain of \( \ln\left(\left|6 + t + t^3\right|\right) \)

The domain is all \( t \) except for the roots of \( 6 + t + t^3 \). Let’s find these roots by solving:

\[
6 + t + t^3 = 0
\]

This is a cubic equation, which typically requires numerical or algebraic methods to find the roots.

### Differentiation of \( \ln\left(\left|6 + t + t^3\right|\right) \)

If you are interested in the derivative of the expression, we apply the chain rule. The derivative of \( \ln\left|u\right| \) with respect to \( u \) is given by:

\[
\frac{d}{dt} \ln\left(\left|6 + t + t^3\right|\right) = \frac{1}{6 + t + t^3} \cdot \frac{d}{dt}(6 + t + t^3)
\]

The derivative of \( 6 + t + t^3 \) is:

\[
\frac{d}{dt}(6 + t + t^3) = 1 + 3t^2
\]

Thus, the derivative becomes:

\[
\frac{1 + 3t^2}{6 + t + t^3}
\]

### Questions You Might Consider Next
1. Would you like a more detailed exploration of the cubic equation \( 6 + t + t^3 = 0 \) and its roots?
2. Are you interested in the second derivative of this logarithmic function?
3. Do you want a graphical representation of \( \ln\left(\left|6 + t + t^3\right|\right) \)?
4. Should we analyze the critical points or inflection points of this function?
5. Would you like help understanding why the logarithm function requires a positive input?

**Tip**: When dealing with logarithmic functions involving absolute values, always remember to carefully consider the domain restrictions imposed by the logarithm.

Explore the logarithmic function ln(|6 + t + t^3|) with a detailed breakdown. Learn about its domain, differentiation, and how to apply the chain rule for derivatives. This problem involves algebraic concepts like cubic equations and logarithms, suitable for students in Grades 10-12.

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