Solve 9x^6+12x^5+4x^4+45x^2+60x+20

$$9 x ^ { 6 } + 12 x ^ { 5 } + 4 x ^ { 4 } + 45 x ^ { 2 } + 60 x + 20$$

Factor

$$\left(3x+2\right)^{2}\left(x^{4}+5\right)$$

Solution Steps

Do the grouping 9x^{6}+12x^{5}+4x^{4}+45x^{2}+60x+20=\left(9x^{6}+12x^{5}+4x^{4}\right)+\left(45x^{2}+60x+20\right), and factor out x^{4} in the first and 5 in the second group.

$$x^{4}\left(9x^{2}+12x+4\right)+5\left(9x^{2}+12x+4\right)$$

Factor out common term 9x^{2}+12x+4 by using distributive property.

$$\left(9x^{2}+12x+4\right)\left(x^{4}+5\right)$$

Consider 9x^{2}+12x+4. Use the perfect square formula, a^{2}+2ab+b^{2}=\left(a+b\right)^{2}, where a=3x and b=2.

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

Rewrite the complete factored expression. Polynomial x^{4}+5 is not factored since it does not have any rational roots.

$$\left(x^{4}+5\right)\left(3x+2\right)^{2}$$

Evaluate

$$\left(3x+2\right)^{2}\left(x^{4}+5\right)$$