Quadratic Inequality Analysis: When is (x+6)(x-3) Greater Than Zero?

To solve this problem, we need to determine when the function $y = (x+6)(x-3)$ is greater than zero. This function is a product of two linear factors, so we will identify for which intervals the product is positive.

First, determine the roots of the function:

• $x+6=0$ gives $x = -6$
• $x-3=0$ gives $x = 3$

The roots divide the number line into three intervals: $x < -6$, $-6 < x < 3$, and $x > 3$.

Next, we test the sign of the product $(x+6)(x-3)$ in each interval:

• For $x < -6$:
  Both factors $(x+6)$ and $(x-3)$ are negative, thus their product is positive. However, actually both factors are negative making their product positive.
• For $-6 < x < 3$:
  The factor $(x+6)$ is positive and $(x-3)$ is negative, thus their product is negative.
• For $x > 3$:
  Both factors $(x+6)$ and $(x-3)$ are positive, thus their product is positive.

Therefore, the function is positive in the intervals $x < -6$ and $x > 3$. Therefore, the correct intervals where $f(x) > 0$ are $x < -6$ and $x > 3$. Based on the choices, the correct answer can be formulated as $x > 3$ or $x < -6$.

However, checking this against the predetermined answer, it appears there may have been an error in the original answer provided. The analysis above suggests choice 4 may have been expected, rather than choice 3. But if we reconsider based on factors again it could be choice 3.

The correct choice, conflicting with what was predetermined, would be actually choice 4.