Solve the Inequality for x y<3x^2+3x-6

$y < 3 x^{2} + 3 x - 6$

Rewrite so $x$ is on the left side of the inequality.

$3 x^{2} + 3 x - 6 > y$

Subtract $y$ from both sides of the inequality.

$3 x^{2} + 3 x - 6 - y > 0$

Convert the inequality to an equation.

$3 x^{2} + 3 x - 6 - y = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 3$ , $b = 3$ , and $c = - 6 - y$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (3 \cdot (- 6 - y))}}{2 \cdot 3}$

Simplify.

Simplify the numerator.

Factor $3$ out of $3^{2} - 4 \cdot 3 \cdot (- 6 - y)$ .

Factor $3$ out of $3^{2}$ .

$x = \frac{- 3 \pm \sqrt{3 \cdot 3 - 4 \cdot 3 \cdot (- 6 - y)}}{2 \cdot 3}$

Factor $3$ out of $- 4 \cdot 3 \cdot (- 6 - y)$ .

$x = \frac{- 3 \pm \sqrt{3 \cdot 3 + 3 (- 4 \cdot (- 6 - y))}}{2 \cdot 3}$

Factor $3$ out of $3 \cdot 3 + 3 (- 4 \cdot (- 6 - y))$ .

$x = \frac{- 3 \pm \sqrt{3 (3 - 4 \cdot (- 6 - y))}}{2 \cdot 3}$

Apply the distributive property.

$x = \frac{- 3 \pm \sqrt{3 (3 - 4 \cdot - 6 - 4 (- y))}}{2 \cdot 3}$

Multiply $- 4$ by $- 6$ .

$x = \frac{- 3 \pm \sqrt{3 (3 + 24 - 4 (- y))}}{2 \cdot 3}$

Multiply $- 1$ by $- 4$ .

$x = \frac{- 3 \pm \sqrt{3 (3 + 24 + 4 y)}}{2 \cdot 3}$

Add $3$ and $24$ .

$x = \frac{- 3 \pm \sqrt{3 (27 + 4 y)}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{- 3 \pm \sqrt{3 (27 + 4 y)}}{6}$

Consolidate the solutions.

$x = - \frac{3 - \sqrt{3 (27 + 4 y)}}{6}$

$x = - \frac{3 + \sqrt{3 (4 y + 27)}}{6}$