Solve 100-4*(6-3m)*(28+9m)>0

$$100 - 4 \times ( 6 - 3 m ) \times ( 28 + 9 m ) > 0$$

Solve for m

$$m\in \left(-\infty,\frac{-\sqrt{454}-5}{9}\right)\cup \left(\frac{\sqrt{454}-5}{9},\infty\right)$$

Steps Using the Quadratic Formula

Multiply -1 and 4 to get -4.

$$100-4\left(6-3m\right)\left(28+9m\right)>0$$

Use the distributive property to multiply -4 by 6-3m.

$$100+\left(-24+12m\right)\left(28+9m\right)>0$$

Use the distributive property to multiply -24+12m by 28+9m and combine like terms.

$$100-672+120m+108m^{2}>0$$

Subtract 672 from 100 to get -572.

$$-572+120m+108m^{2}>0$$

To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-572+120m+108m^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 108 for a, 120 for b, and -572 for c in the quadratic formula.

$$m=\frac{-120±\sqrt{120^{2}-4\times 108\left(-572\right)}}{2\times 108}$$

Do the calculations.

$$m=\frac{-120±24\sqrt{454}}{216}$$

Solve the equation m=\frac{-120±24\sqrt{454}}{216} when ± is plus and when ± is minus.

$$m=\frac{\sqrt{454}-5}{9} m=\frac{-\sqrt{454}-5}{9}$$

Rewrite the inequality by using the obtained solutions.

$$108\left(m-\frac{\sqrt{454}-5}{9}\right)\left(m-\frac{-\sqrt{454}-5}{9}\right)>0$$

For the product to be positive, m-\frac{\sqrt{454}-5}{9} and m-\frac{-\sqrt{454}-5}{9} have to be both negative or both positive. Consider the case when m-\frac{\sqrt{454}-5}{9} and m-\frac{-\sqrt{454}-5}{9} are both negative.

$$m-\frac{\sqrt{454}-5}{9}<0 m-\frac{-\sqrt{454}-5}{9}<0$$

The solution satisfying both inequalities is m<\frac{-\sqrt{454}-5}{9}.

$$m<\frac{-\sqrt{454}-5}{9}$$

Consider the case when m-\frac{\sqrt{454}-5}{9} and m-\frac{-\sqrt{454}-5}{9} are both positive.

$$m-\frac{-\sqrt{454}-5}{9}>0 m-\frac{\sqrt{454}-5}{9}>0$$

The solution satisfying both inequalities is m>\frac{\sqrt{454}-5}{9}.

$$m>\frac{\sqrt{454}-5}{9}$$

The final solution is the union of the obtained solutions.

$$m<\frac{-\sqrt{454}-5}{9}\text{; }m>\frac{\sqrt{454}-5}{9}$$