+1418 Find all $x$ that satisfy the inequality $(2x+10)(x+3)<(3x+1)(x+6)$. Express your answer in interval notation.
+1624 Expanding: \(2x^2+16x+30<3x^2+19x+6\)
Combine like terms: \(x^2+3x-24>0\)
Quadratic formula: \(x={-3\pm\sqrt{105}\over2}\), which are the two roots r and s, used in the factoring of the quadratic (x - r)(x - s) = x^2 + 3x - 24.
To be positive, we take the extreme values in which both (x - r) and (x - s) are positive or both negative, so our solution interval is:
\((-\inf, {-3-\sqrt{105}\over2})\) U \(({-3 + \sqrt{105}\over2}, \inf)\)
