To simplify the expression ((2x + 3) \cdot |x + 1|), we need to consider the properties of absolute values, as they impact the expression depending on the value of (x).
Case 1: (x + 1 \geq 0) (which means (x \geq -1))
In this case, (|x + 1| = x + 1). Therefore, the expression becomes:
[
(2x + 3) \cdot |x + 1| = (2x + 3) \cdot (x + 1)
]
Now we can distribute:
[
= 2x^2 + 2x + 3x + 3 = 2x^2 + 5x + 3
]
Case 2: (x + 1 < 0) (which means (x < -1))
Here, (|x + 1| = -(x + 1)). Thus, the expression becomes:
[
(2x + 3) \cdot |x + 1| = (2x + 3) \cdot (-(x + 1))
]
Distributing this, we have:
[
= - (2x + 3)(x + 1) = - (2x^2 + 2x + 3x + 3) = - (2x^2 + 5x + 3) = -2x^2 - 5x - 3
]
Summary of the Results
The expression ( (2x + 3) \cdot |x + 1| ) can be summarized as follows:
[
(2x + 3) \cdot |x + 1| =
\begin{cases}
2x^2 + 5x + 3 & \text{if } x \geq -1 \
-2x^2 - 5x - 3 & \text{if } x < -1
\end{cases}
]
