Let's first analyze what the problem is asking.

We are to find the value of the expression:

$$3\sin^2{\alpha} + 10 + 3\cos^2{\alpha}$$

Let’s develop a step-by-step plan to solve it:

Step 1: Notice the Pythagorean identity involving the sine and cosine functions:

$$\sin^2{\alpha} + \cos^2{\alpha} = 1$$

This will allow us to combine like terms.

Step 2: Rewrite the original expression by grouping the sine and cosine terms together:

$$3\sin^2{\alpha} + 3\cos^2{\alpha} + 10$$

Step 3: Factor out the common factor of 3:

$3(\sin^2{\alpha} + \cos^2{\alpha}) + 10$

Step 4: Use the Pythagorean identity to substitute $\sin^2{\alpha} + \cos^2{\alpha}$ with 1:

$$3 \times 1 + 10$$

Step 5: Calculate the final value:

$$3 + 10 = 13$$

Now, let's confirm our computation. The value of the expression is:

$3\sin^2{\alpha} + 10 + 3\cos^2{\alpha} = 3(\sin^2{\alpha} + \cos^2{\alpha}) + 10 = 3 \times 1 + 10 = 13$

The answer is $13$.