Question 1033449
Most of the work below is wrong having a mistake; I might fix and repost later:

{{{sqrt(2x+1)+5=sqrt(x+12)-8}}}
{{{sqrt(2x+1)=sqrt(x+12)-8-5}}}
{{{sqrt(2x+1)=sqrt(x+12)-13}}}
square both sides....
{{{2x+1=(x+12)-26*sqrt(x+12)+169}}}
{{{2x+1-x-12-169=-26*sqrt(x+12)}}}
{{{-2x-1+x+12+169=26*sqrt(x+12)}}}
{{{-x+11+169=26sqrt(x+12)}}}
{{{180-x=26*sqrt(x+12)}}}
Square both sides AGAIN...
{{{676(x+12)=180^2-360x+x^2}}}
{{{676x+676*12=x^2-360x+180^2}}}
{{{0=x^2-360+180^2-676x-676*12}}}
{{{x^2-676x-360+180^2-676*12=0}}}
{{{highlight_green(x^2-676x+23928=0)}}}


You might need to check if one or both of those solutions work in the original equation.


General Solution Formula for Quadratic Equation:
{{{x=(676+- sqrt(676^2-4*23928))/2}}}
{{{x=(676+- sqrt(361264))/2}}}
-
D=16*22579=16*67*337
{{{x=(676+- sqrt(16*67*337))/2}}}
{{{x=(676+- 4*sqrt(67*337))/2}}}
{{{highlight(cross(x=338+- 2*sqrt(67*337)))}}}   ---------check to be sure which of these work in case only one.



*That solution is wrong.  The correct results, not yet both checked, are x=24 or x=1012.