To simplify this expression, we will first calculate the numerator and denominator separately.

Numerator:
cos(10°) + sin(40°) + 3√2

Using trigonometric identities:
cos(10°) = cos(40°) = sin(50°) (since cos(x) = sin(90° - x))

Therefore,
cos(10°) + sin(40°) = sin(50°) + sin(40°)
= 2cos(5°)sin(45°)
= 2cos(5°)(√2/2)
= cos(5°)√2

Now, the numerator becomes:
cos(5°)√2 + 3√2
= (√2)(cos(5°) + 3)
= (√2)(cos(85°) + 3) (since cos(85°) = sin(5°))

Denominator:
cos(40°) + sin(10°) + √6

Using trigonometric identities:
cos(40°) = sin(50°)

Therefore,
cos(40°) + sin(10°) = sin(50°) + sin(10°)
= 2cos(20°)sin(30°)
= 2(√3/2)(1/2)
= √3

Now, the denominator becomes:
√3 + √6

Now, the simplified expression is:
(√2)(cos(85°) + 3) / (√3 + √6)