#1**0 **

To solve the inequality \(x(x + 6) > 16\), follow these steps:

1. Expand the left side of the inequality:

\(x^2 + 6x > 16\)

2. Move all terms to one side of the inequality:

\(x^2 + 6x - 16 > 0\)

3. Factor the quadratic expression:

\((x + 8)(x - 2) < 0\)

Now we have the factors of the quadratic expression. To determine the intervals where the inequality is satisfied, you can use a sign chart or test points within the intervals. The inequality is satisfied when the expression is greater than 0. Let's analyze the sign of the expression within different intervals:

- When \(x < -8\):

Both factors \((x + 8)\) and \((x - 2)\) are negative. Their product is positive. So, the inequality holds in this interval.

- When \(-8 < x < 2\):

\((x + 8)\) is positive, and \((x - 2)\) is negative. Their product is negative. So, the inequality holds in this interval.

- When \(x > 2\):

Both factors \((x + 8)\) and \((x - 2)\) are positive. Their product is negative. So, the inequality does not hold in this interval.

Now, we can write the solution in interval notation:

Solution: \((- \infty, -8) \cup (-8,2)\)

This indicates that the inequality \(x(x + 6) > 16\) is satisfied when \(x\) is in the intervals from negative infinity to -8 (excluding -8), and from -8 (excluding -8) to 2 (excluding 2).

SpectraSynth Aug 17, 2023