The equation provided in the image is:

$\sqrt{3 - x} - \sqrt{7 + x} = \sqrt{16 + 2x}$

To solve this equation, we'll proceed step by step.

Step 1: Isolate one of the square roots

First, let's try to isolate one of the square roots. We can start by isolating $\sqrt{16 + 2x}$:

$\sqrt{3 - x} - \sqrt{7 + x} = \sqrt{16 + 2x}$

Add $\sqrt{7 + x}$ to both sides:

$\sqrt{3 - x} = \sqrt{7 + x} + \sqrt{16 + 2x}$

Step 2: Square both sides

Next, square both sides of the equation to eliminate the square roots:

$(\sqrt{3 - x})^2 = (\sqrt{7 + x} + \sqrt{16 + 2x})^2$

On the left side:

$3 - x$

On the right side, use the identity $(a + b)^2 = a^2 + 2ab + b^2$:

$7 + x + 16 + 2x + 2\sqrt{(7 + x)(16 + 2x)}$

Simplifying:

$3 - x = 23 + 3x + 2\sqrt{(7 + x)(16 + 2x)}$

Step 3: Isolate the square root again

Now, isolate the square root term:

$3 - x - 23 - 3x = 2\sqrt{(7 + x)(16 + 2x)}$

Simplify:

$-4x - 20 = 2\sqrt{(7 + x)(16 + 2x)}$

Divide by 2:

$-2x - 10 = \sqrt{(7 + x)(16 + 2x)}$

Step 4: Square both sides again

Square both sides again:

$(-2x - 10)^2 = (7 + x)(16 + 2x)$

Expand both sides:

$4x^2 + 40x + 100 = 112 + 30x + 2x^2$

Step 5: Simplify and solve the quadratic equation

Move all terms to one side:

$4x^2 + 40x + 100 - 112 - 30x - 2x^2 = 0$

Simplify:

$2x^2 + 10x - 12 = 0$

Divide by 2:

$x^2 + 5x - 6 = 0$

Factor the quadratic:

$(x + 6)(x - 1) = 0$

So, the solutions are:

$x = -6 \quad \text{or} \quad x = 1$

Step 6: Verify the solutions

We need to check both potential solutions in the original equation.

For $x = -6$:

$\sqrt{3 - (-6)} - \sqrt{7 + (-6)} = \sqrt{16 + 2(-6)}$

$\sqrt{9} - \sqrt{1} = \sqrt{4}$

$3 - 1 = 2 \quad \text{which is true}$

For $x = 1$:

[ \sqrt{3 -