Mathematics

This problem hinges on recognizing that both logarithmic expressions can be simplified by factoring their arguments. Notice that $6x^2 + 28x + 30$ and $x^2 + 6x + 9$ are perfect squares or products of linear factors, and the bases $x+3$ and $6x+10$ are related to those factors. Once you rewrite each logarithm in terms of simpler bases, the equation collapses into something much more manageable—often a polynomial you can solve directly. The key is careful algebraic manipulation before attempting to isolate $x$.

This problem hinges on recognizing that both logarithmic expressions can be simplified by factoring their arguments. Notice that $6x^2 + 28x + 30$ and $x^2 + 6x + 9$ are perfect squares or products of linear factors, and the bases $x+3$ and $6x+10$ are related to those factors. Once you rewrite each logarithm in terms of simpler bases, the equation collapses into something much more manageable—often a polynomial you can solve directly. The key is careful algebraic manipulation before attempting to isolate $x$.