Solve -6x^6-12x^5+210x^4

$$- 6 x ^ { 6 } - 12 x ^ { 5 } + 210 x ^ { 4 }$$

Factor

$$6\left(5-x\right)\left(x+7\right)x^{4}$$

Solution Steps

Factor out 6.

$$6\left(-x^{6}-2x^{5}+35x^{4}\right)$$

Consider -x^{6}-2x^{5}+35x^{4}. Factor out x^{4}.

$$x^{4}\left(-x^{2}-2x+35\right)$$

Consider -x^{2}-2x+35. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+35. To find a and b, set up a system to be solved.

$$a+b=-2 ab=-35=-35$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -35.

$$1,-35 5,-7$$

Calculate the sum for each pair.

$$1-35=-34 5-7=-2$$

The solution is the pair that gives sum -2.

$$a=5 b=-7$$

Rewrite -x^{2}-2x+35 as \left(-x^{2}+5x\right)+\left(-7x+35\right).

$$\left(-x^{2}+5x\right)+\left(-7x+35\right)$$

Factor out x in the first and 7 in the second group.

$$x\left(-x+5\right)+7\left(-x+5\right)$$

Factor out common term -x+5 by using distributive property.

$$\left(-x+5\right)\left(x+7\right)$$

Rewrite the complete factored expression.

$$6x^{4}\left(-x+5\right)\left(x+7\right)$$

Evaluate

$$6\left(5-x\right)\left(x+7\right)x^{4}$$