Solve 2x^2+16x+32+12x^3+48x^2-4x-16

$$2 x ^ { 2 } + 16 x + 32 + 12 x ^ { 3 } + 48 x ^ { 2 } - 4 x - 16$$

Evaluate

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

Solution Steps

Combine 2x^{2} and 48x^{2} to get 50x^{2}.

$$50x^{2}+16x+32+12x^{3}-4x-16$$

Combine 16x and -4x to get 12x.

$$50x^{2}+12x+32+12x^{3}-16$$

Subtract 16 from 32 to get 16.

$$50x^{2}+12x+16+12x^{3}$$

Factor

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

Solution Steps

Factor out 2.

$$2\left(25x^{2}+6x+8+6x^{3}\right)$$

Consider x^{2}+8x+16+6x^{3}+24x^{2}-2x-8. Multiply and combine like terms.

$$6x^{3}+25x^{2}+6x+8$$

Consider 6x^{3}+25x^{2}+6x+8. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 8 and q divides the leading coefficient 6. One such root is -4. Factor the polynomial by dividing it by x+4.

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

Rewrite the complete factored expression. Polynomial 6x^{2}+x+2 is not factored since it does not have any rational roots.

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