Solve 6x^4+24x^3-360x^2

$$6 x ^ { 4 } + 24 x ^ { 3 } - 360 x ^ { 2 }$$

Factor

$$6\left(x-6\right)\left(x+10\right)x^{2}$$

Solution Steps

Factor out 6.

$$6\left(x^{4}+4x^{3}-60x^{2}\right)$$

Consider x^{4}+4x^{3}-60x^{2}. Factor out x^{2}.

$$x^{2}\left(x^{2}+4x-60\right)$$

Consider x^{2}+4x-60. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-60. To find a and b, set up a system to be solved.

$$a+b=4 ab=1\left(-60\right)=-60$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -60.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Calculate the sum for each pair.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

The solution is the pair that gives sum 4.

$$a=-6 b=10$$

Rewrite x^{2}+4x-60 as \left(x^{2}-6x\right)+\left(10x-60\right).

$$\left(x^{2}-6x\right)+\left(10x-60\right)$$

Factor out x in the first and 10 in the second group.

$$x\left(x-6\right)+10\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(x+10\right)$$

Rewrite the complete factored expression.

$$6x^{2}\left(x-6\right)\left(x+10\right)$$

Evaluate

$$6\left(x-6\right)\left(x+10\right)x^{2}$$