Let's solve the given equation step-by-step. The equation is:

$5(3x + 5) = 6(4x + 5)$

Step 1: Distribute both sides of the equation.

On the left side: $5(3x + 5) = 5 \cdot 3x + 5 \cdot 5 = 15x + 25$

On the right side: $6(4x + 5) = 6 \cdot 4x + 6 \cdot 5 = 24x + 30$

Now the equation becomes: $15x + 25 = 24x + 30$

Step 2: Rearrange the equation to get all terms involving $x$ on one side and the constant terms on the other side.

First, subtract $15x$ from both sides: $25 = 9x + 30$

Next, subtract 30 from both sides: $25 - 30 = 9x$ $-5 = 9x$

Step 3: Solve for $x$.

Divide both sides by 9: $x = \frac{-5}{9}$

Thus, the solution is: $x = \frac{-5}{9}$

Would you like any further explanation?

Here are five related questions to explore:

1. How would the solution change if the equation were $5(3x + 7) = 6(4x + 5)$?
2. Can you solve a similar equation involving a negative coefficient, such as $7(3x - 4) = 5(2x + 8)$?
3. What happens if you multiply both sides of the equation by a constant before distributing?
4. How can you check if your solution to a linear equation is correct?
5. What is the general process for solving linear equations that involve fractions?

Tip: Always simplify both sides of the equation before moving terms across the equals sign to avoid mistakes.