Question 173614
Add {{{sqrt (x + 1)}}} to each side of the equation
{{{sqrt (2x + 3) - sqrt (x + 1)+ sqrt (x + 1) = 1 + sqrt (x + 1)}}}
{{{sqrt (2x + 3) = 1 + sqrt (x + 1)}}}
Square each side of the equation
{{{(2x + 3) = 1 + 2*sqrt (x + 1) + (x+1)}}}
{{{2x + 3 - 1 - (x + 1) = 1 + 2*sqrt (x + 1) + (x+1) - 1 -(x+1)}}}
{{{x + 1  = 2*sqrt (x + 1)}}}
Square each side of the equation again
{{{x^2 + 2x + 1 = 4*(x + 1)}}}
{{{x^2 + 2x + 1 = 4x + 4}}}
{{{x^2 - 2x - 3 = 0}}}
{{{(x + 1) (x - 3) = 0}}}
ie x = -1 or x = 3
Substitute back into the original equation and test if they both work.