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

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

Factor

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

Solution Steps

Factor out 3.

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

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

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

Consider -x^{2}+x+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.

$$a+b=1 ab=-20=-20$$

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 -20.

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

Calculate the sum for each pair.

$$-1+20=19 -2+10=8 -4+5=1$$

The solution is the pair that gives sum 1.

$$a=5 b=-4$$

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

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

Factor out -x in the first and -4 in the second group.

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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