Find Intervals of Increase and Decrease: y = x² + 5x + 4

The problem asks us to determine the intervals where the function $y = x^2 + 5x + 4$ is increasing and where it is decreasing.

Let's analyze this systematically using the following steps:

• Step 1: Identify Key Information
  We have the function $y = x^2 + 5x + 4$. This is a quadratic function, represented in the standard form $ax^2 + bx + c$, where $a=1$, $b=5$, and $c=4$.
• Step 2: Determine the Vertex
  The vertex of a parabola described by a quadratic function $ax^2 + bx + c$ is given by the formula $x = -\frac{b}{2a}$. Substituting $a=1$ and $b=5$, we get:
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$x = -\frac{5}{2 \times 1} = -\frac{5}{2} = -2.5$

• Step 3: Use Vertex to Find Critical Point
  The vertex divides the parabola into two distinct sections: one that increases and one that decreases. The point $x = -2.5$ is the critical point where the transition occurs between decreasing and increasing intervals.
• Step 4: Determine the Intervals
  Since the leading coefficient $a = 1$ is positive, the parabola opens upwards. Consequently, the function decreases to the left of the vertex and increases to the right of the vertex.

Therefore, the intervals are:
- Decreasing: $x < -2.5$
- Increasing: $x > -2.5$

In conclusion, the solution to the problem is:

$\searrow : x < -2.5$ (function decreases)
$\nearrow : x > -2.5$ (function increases)