Solve 3x^4+27x^3+60x^2

$$3 x ^ { 4 } + 27 x ^ { 3 } + 60 x ^ { 2 }$$

Factor

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

Solution Steps

Factor out 3.

$$3\left(x^{4}+9x^{3}+20x^{2}\right)$$

Consider x^{4}+9x^{3}+20x^{2}. Factor out x^{2}.

$$x^{2}\left(x^{2}+9x+20\right)$$

Consider x^{2}+9x+20. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+20. To find a and b, set up a system to be solved.

$$a+b=9 ab=1\times 20=20$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 20.

$$1,20 2,10 4,5$$

Calculate the sum for each pair.

$$1+20=21 2+10=12 4+5=9$$

The solution is the pair that gives sum 9.

$$a=4 b=5$$

Rewrite x^{2}+9x+20 as \left(x^{2}+4x\right)+\left(5x+20\right).

$$\left(x^{2}+4x\right)+\left(5x+20\right)$$

Factor out x in the first and 5 in the second group.

$$x\left(x+4\right)+5\left(x+4\right)$$

Factor out common term x+4 by using distributive property.

$$\left(x+4\right)\left(x+5\right)$$

Rewrite the complete factored expression.

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

Evaluate

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