We are given the following equation:
[tex]\frac{-4}{x+4}=-\frac{3}{x+6}[/tex]we are asked to determine any extraneous solutions. To do that we will determine the values of "x" that solve the equation. First, we will cross multiply the equation:
[tex]-4(x+6)=-3(x+4)[/tex]Now, we can multiply by -1, we get:
[tex]4(x+6)=3(x+4)[/tex]Now we use the distributive property on both sides, we get:
[tex]4x+24=3x+12[/tex]Now, we subtract "3x" from both sides:
[tex]\begin{gathered} 4x-3x+24=3x-3x+12 \\ x+24=12 \end{gathered}[/tex]Now we subtract 24 from both sides:
[tex]\begin{gathered} x+24-24=12-24 \\ x=-12 \end{gathered}[/tex]Therefore, the solution is x = -12. The extraneous solutions are the solutions that are not in the domain of the original function.
In the domain of the original function, we have that the values of:
[tex]\begin{gathered} x=-4 \\ x=-6 \end{gathered}[/tex]Would make the denominators equal to zero, and therefore, they are not in the domain. Since the solution is none of these values there are no extraneous solutions.