Solve 6x^3+12x^2-90x-216

$$6 x ^ { 3 } + 12 x ^ { 2 } - 90 x - 216$$

Factor

$$6\left(x-4\right)\left(x+3\right)^{2}$$

Solution Steps

Factor out 6.

$$6\left(x^{3}+2x^{2}-15x-36\right)$$

Consider x^{3}+2x^{2}-15x-36. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -36 and q divides the leading coefficient 1. One such root is 4. Factor the polynomial by dividing it by x-4.

$$\left(x-4\right)\left(x^{2}+6x+9\right)$$

Consider x^{2}+6x+9. Use the perfect square formula, a^{2}+2ab+b^{2}=\left(a+b\right)^{2}, where a=x and b=3.

$$\left(x+3\right)^{2}$$

Rewrite the complete factored expression.

$$6\left(x-4\right)\left(x+3\right)^{2}$$

Evaluate

$$6\left(x-4\right)\left(x+3\right)^{2}$$