To solve this inequality, we need to first combine the logarithms on the left side using the properties of logarithms.

log √2(x+5) + log √2(4-x) = log √2[(x+5)(4-x)]

Now, we can rewrite the inequality in a simplified form:

log √2[(x+5)(4-x)] > log √2(5-3x)

Next, we can remove the logarithms by raising both sides to the power of √2:

[(x+5)(4-x)] > (5-3x)

Expand the left side:

4x - x^2 + 20 - 5x > 5 - 3x

Simplify and rearrange the terms:

-x^2 - 4x + 20 > -3x + 5

Add 3x and subtract 5 from both sides:

-x^2 - x + 15 > 0

Now, the inequality is in the form of a quadratic equation. To solve this quadratic inequality, we can factor or use the quadratic formula:

The quadratic inequality -x^2 - x + 15 > 0 can be factored as (-x + 5)(x + 3) > 0

The sign analysis of the inequality shows that the solution is x ∈ (-∞, -3) U (5, ∞)

Therefore, the solution to the inequality log √2(x+5) + log √2(4-x) > log √2(5-3x) is x ∈ (-∞, -3) U (5, ∞)