What would you like to understand?

It is valid to divide by $e^{rt}$ because the exponential function is never zero for any real values of $r$ and $t$. Since the factor is strictly nonzero, dividing both sides of the equation by it does not lose any solutions for $r$, allowing the differential equation to reduce to the algebraic characteristic equation.

Conditions: The equation is of the form $0 = e^{rt}[ar^2 + br + c]$.; $r$ and $t$ are real numbers.

It is valid to divide by $e^{rt}$ because the exponential function is never zero for any real values of $r$ and $t$. Since the factor is strictly nonzero, dividing both sides of the equation by it does not lose any solutions for $r$, allowing the differential equation to reduce to the algebraic characteristic equation.

Conditions: The equation is of the form $0 = e^{rt}[ar^2 + br + c]$.; $r$ and $t$ are real numbers.