Answer
$\{-3\}$.
Work Step by Step
Substitute $\frac{4}{x-1}=u$ in the given equation.
$\Rightarrow \left (u \right )^2+2\left (u \right )+1=0$
$\Rightarrow u ^2+2u +1=0$
Use the special formula $(a+b)^2=a^2+2ab+b^2$.
$\Rightarrow (u+1)^2=0$
By using zero product rule set the factor equal to zero.
$\Rightarrow u+1=0$
Isolate $u$.
$\Rightarrow u=-1$.
Back substitute $u=\frac{4}{x-1}$.
$\Rightarrow \frac{4}{x-1}=-1$.
Multiply the equation by $(x-1)$ to clear the denominator
$\Rightarrow (x-1)\cdot \frac{4}{x-1}=(x-1)\cdot(-1)$.
Simplify.
$\Rightarrow 4=1-x$.
Add $-4+x$ to both sides.
$\Rightarrow 4-4+x=1-x-4+x$.
Simplify.
$\Rightarrow x=-3$.
The solution set is $\{-3\}$.
Note: Check if the solution is correct. The equation is defined for all real values of $x$ except the zeros of the denominators, which is $x=1$. Since $-3\not =1$, the solution is correct.
