Hi AAPL.

Let's take a look at your question.

Let's try option by option:

(1) x-4 = -11, this implies that x=-7. Then $$\sqrt{\left(x+4\right)^2}=\sqrt{\left(-7+4\right)^2}=\sqrt{\left(-3\right)^2}=3.\ \ \text{TRUE}$$

(2) x-4 = -7 this implies that x=-3. Then $$\sqrt{\left(x+4\right)^2}=\sqrt{\left(-3+4\right)^2}=\sqrt{\left(1\right)^2}=1\ne3.\ \ \text{FALSE}$$

(3) x-4 = -4 this implies that x=0. Then $$\sqrt{\left(x+4\right)^2}=\sqrt{\left(0+4\right)^2}=\sqrt{\left(4\right)^2}=4\ne3.\ \ \text{FALSE}$$

(4) x-4 = -3 this implies that x=1. Then $$\sqrt{\left(x+4\right)^2}=\sqrt{\left(1+4\right)^2}=\sqrt{\left(5\right)^2}=5\ne3.\ \ \text{FALSE}$$

(5) x-4 = 5 this implies that x=9. Then $$\sqrt{\left(x+4\right)^2}=\sqrt{\left(9+4\right)^2}=\sqrt{\left(13\right)^2}=13\ne3.\ \ \text{FALSE}$$

Hence, the correct answer is the option A.

I hope this answer can help you.

Feel free to ask me again if you have a doubt.

Regards.