The problem is to simplify the expression $\sin^2{\alpha} + \sin{\alpha}\cos^2{\alpha}$.

1. Problem Description

The problem is to simplify the expression $\sin^2{\alpha} + \sin{\alpha}\cos^2{\alpha}$.

2. Solution Steps

First, we can factor out $\sin{\alpha}$ from the expression:

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

Since we know that $\sin^2{\alpha} + \cos^2{\alpha} = 1$, we can write $\sin{\alpha} = 1 - \cos^2{\alpha}$.

Now, substitute $\sin{\alpha} = 1 - \cos^2{\alpha}$ into the factored expression:

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

Alternatively, factor out $\sin{\alpha}$ to get:

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

We know that $\sin^2{\alpha} + \cos^2{\alpha} = 1$. Thus $\sin{\alpha} = 1 - \cos^2{\alpha}$.

Then, we can write:

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

Let's go back to $\sin^2{\alpha} + \sin{\alpha}\cos^2{\alpha}$. We can factor out $\sin{\alpha}$.

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

Then $\sin{\alpha}(\sin{\alpha} + \cos^2{\alpha}) = \sin{\alpha}(\sin{\alpha} + 1 - \sin^2{\alpha})$.

Let's rewrite the expression as $\sin{\alpha}(\sin{\alpha} + \cos^2{\alpha})$.

We can also write this as $\sin{\alpha}(\sin{\alpha} + \cos^2{\alpha}) = \sin{\alpha}(\sin{\alpha} + 1 - \sin^2{\alpha})$.

Thus, $\sin{\alpha}(\sin{\alpha} + \cos^2{\alpha}) = \sin{\alpha}$.

3. Final Answer

$\sin{\alpha}$