To solve the equation:
[tex]\[
\frac{4}{x-5} = \sqrt{x+3} + 2
\][/tex]
we can follow these steps:
### Step 1: Rearrange the equation
Firstly, we want to isolate terms involving [tex]\( x \)[/tex]. The given equation can be written without changing the structure:
[tex]\[
\frac{4}{x-5} = \sqrt{x+3} + 2
\][/tex]
### Step 2: Consider possible values
Replace [tex]\( x \)[/tex] with the values given in the options to check which one satisfies the equation approximately.
#### Option A: [tex]\( x \approx 3.73 \)[/tex]
[tex]\[
\frac{4}{3.73-5} = \sqrt{3.73+3} + 2
\][/tex]
[tex]\[
\frac{4}{-1.27} \approx \sqrt{6.73} + 2
\][/tex]
[tex]\[
\approx -3.15 \neq 4.60
\][/tex]
This value does not satisfy the equation.
#### Option B: [tex]\( x \approx 4.97 \)[/tex]
[tex]\[
\frac{4}{4.97-5} = \sqrt{4.97+3} + 2
\][/tex]
[tex]\[
\frac{4}{-0.03} \approx \sqrt{7.97} + 2
\][/tex]
[tex]\[
\approx -133.33 \neq 4.82
\][/tex]
This value also does not satisfy the equation.
#### Option C: [tex]\( x \approx -0.80 \)[/tex]
[tex]\[
\frac{4}{-0.80-5} = \sqrt{-0.80+3} + 2
\][/tex]
[tex]\[
\frac{4}{-5.80} \approx \sqrt{2.20} + 2
\][/tex]
[tex]\[
\approx -0.69 \neq 3.48
\][/tex]
This value does not satisfy the equation either.
#### Option D: [tex]\( x \approx 5.81 \)[/tex]
[tex]\[
\frac{4}{5.81-5} = \sqrt{5.81+3} + 2
\][/tex]
[tex]\[
\frac{4}{0.81} \approx \sqrt{8.81} + 2
\][/tex]
[tex]\[
\approx 4.94 \approx 4.97
\][/tex]
This value satisfies the equation approximately.
### Conclusion
The correct approximate solution that satisfies the equation is:
[tex]\[
x \approx 5.81
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
