Analyze x² + 10x + 25: Exploring Positive and Negative Domains

To solve this problem, follow these steps:

• Step 1: Factor the quadratic function or recognize it as a perfect square trinomial.
• Step 2: Identify the vertex of the parabola.
• Step 3: Analyze the intervals between roots and the surrounding areas to determine positivity or negativity.

Now, work through these steps:

Step 1: Recognize the quadratic function $y = x^2 + 10x + 25$ as a perfect square trinomial. It can be rewritten as $y = (x + 5)^2$.

Step 2: The vertex of the parabola, which also represents its minimum point since the parabola opens upwards, occurs at $x = -5$.

Step 3: Since $(x + 5)^2 \geq 0$ for all real $x$ (because a square is always non-negative), the function is non-negative everywhere.

Therefore, the function is never negative, and since it equals zero at $x = -5$, it is positive for all $x \neq -5$.

Therefore, the function's positive domain is $x > 0 : x \neq -5$. For $x < 0$, the function is not negative, hence there is no such domain.

The solution to the problem is:

$x > 0 : x \neq -5$

$x < 0 :$ none