Let's check if \( x = -4 \) is a solution by substituting it into the original equation and verifying if both sides are equal.

### Original Equation:
\[
6^{3x + 1} = 2^{2x - 3}
\]

### Step 1: Substitute \( x = -4 \) into the equation

#### Left-hand side:
\[
3x + 1 = 3(-4) + 1 = -12 + 1 = -11
\]
\[
\text{So, } 6^{3x + 1} = 6^{-11}
\]

#### Right-hand side:
\[
2x - 3 = 2(-4) - 3 = -8 - 3 = -11
\]
\[
\text{So, } 2^{2x - 3} = 2^{-11}
\]

### Step 2: Compare both sides

We have:
\[
6^{-11} \quad \text{and} \quad 2^{-11}
\]

Clearly, \( 6^{-11} \) and \( 2^{-11} \) are not equal because 6 and 2 are different bases. 

### Conclusion:
Since \( 6^{-11} \) is not equal to \( 2^{-11} \), \( x = -4 \) is **not** a solution to the equation.

---

Would you like to explore why \( x = -4 \) does not work or have any other questions?

Here are 8 related questions you might consider:
1. What happens when you plug in \( x = -4 \) into both sides?
2. How can you verify if another value of \( x \) is a solution?
3. How would you graph both sides of the equation to find the solution?
4. What is the significance of logarithmic properties in solving equations?
5. How would the solution change if the bases of the exponents were the same?
6. What steps could you take if both sides of the equation were more complex?
7. How can you use numerical methods to approximate solutions?
8. How would you check your work if solving the equation without a calculator?

**Tip:** Always verify potential solutions by substituting them back into the original equation to ensure they satisfy both sides.

Explore whether x = -4 is a solution to the equation 6^(3x + 1) = 2^(2x - 3). Check step-by-step how substituting x = -4 into the equation leads to different bases for comparison, showing why it's not a solution. Suitable for high school students studying exponential equations.

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