Solve 14x^3+28x^2+58x^2+176x+120

$$14 { x }^{ 3 } +28 { x }^{ 2 } +58 { x }^{ 2 } +176x+120$$

Factor

$$2\left(7x+15\right)\left(x+2\right)^{2}$$

Solution Steps

Factor out 2.

$$2\left(7x^{3}+14x^{2}+29x^{2}+88x+60\right)$$

Consider 7x^{3}+14x^{2}+29x^{2}+88x+60. Multiply and combine like terms.

$$7x^{3}+43x^{2}+88x+60$$

Consider 7x^{3}+43x^{2}+88x+60. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 60 and q divides the leading coefficient 7. One such root is -\frac{15}{7}. Factor the polynomial by dividing it by 7x+15.

$$\left(7x+15\right)\left(x^{2}+4x+4\right)$$

Consider x^{2}+4x+4. Use the perfect square formula, a^{2}+2ab+b^{2}=\left(a+b\right)^{2}, where a=x and b=2.

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

Rewrite the complete factored expression.

$$2\left(7x+15\right)\left(x+2\right)^{2}$$

Evaluate

$$2\left(7x+15\right)\left(x+2\right)^{2}$$

Solution Steps

Combine 28x^{2} and 58x^{2} to get 86x^{2}.

$$14x^{3}+86x^{2}+176x+120$$