The problem is to simplify the expression $\sin^2(x) + \sin(x)\cos(x)$.

1. Problem Description

The problem is to simplify the expression $\sin^2(x) + \sin(x)\cos(x)$.

2. Solution Steps

We are given the expression $\sin^2(x) + \sin(x)\cos(x)$.

We can factor out $\sin(x)$ from both terms.

$\sin^2(x) + \sin(x)\cos(x) = \sin(x)(\sin(x) + \cos(x))$

The expression can also be written using the identity $\sin(2x) = 2\sin(x)\cos(x)$

$\sin(x)\cos(x) = \frac{1}{2}\sin(2x)$

Therefore, we have:

$\sin^2(x) + \sin(x)\cos(x) = \sin^2(x) + \frac{1}{2}\sin(2x)$

We can also use the identity $\sin^2(x) = \frac{1 - \cos(2x)}{2}$.

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

$= \frac{1}{2} - \frac{1}{2}\cos(2x) + \frac{1}{2}\sin(2x)$

$= \frac{1}{2} (1 + \sin(2x) - \cos(2x))$

However, $\sin(x)(\sin(x) + \cos(x))$ is the simplest factored form.

3. Final Answer

$\sin(x)(\sin(x) + \cos(x))$