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

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

Alternatively, we can multiply and divide by 2 the second term to try to use trigonometric identities.

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

Using the double angle identity:

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

we get

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

This expression is also a correct simplification.

However, $\sin(x)(\sin(x) + \cos(x))$ seems like the more natural simplification of the original expression.

3. Final Answer

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