Solve -1-3x+4x^2+4x-3x^3-1

$$-1-3x+4 { x }^{ 2 } +4x-3 { x }^{ 3 } -1$$

Evaluate

$$-\left(3x+2\right)\left(1-x\right)^{2}$$

Solution Steps

Combine -3x and 4x to get x.

$$-1+x+4x^{2}-3x^{3}-1$$

Subtract 1 from -1 to get -2.

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

Factor

$$-\left(3x+2\right)\left(1-x\right)^{2}$$

Solution Steps

Multiply and combine like terms.

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

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -2 and q divides the leading coefficient -3. One such root is -\frac{2}{3}. Factor the polynomial by dividing it by 3x+2.

$$\left(3x+2\right)\left(-x^{2}+2x-1\right)$$

Consider -x^{2}+2x-1. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=2 ab=-\left(-1\right)=1$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=1$$

Rewrite -x^{2}+2x-1 as \left(-x^{2}+x\right)+\left(x-1\right).

$$\left(-x^{2}+x\right)+\left(x-1\right)$$

Factor out -x in -x^{2}+x.

$$-x\left(x-1\right)+x-1$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(-x+1\right)$$

Rewrite the complete factored expression.

$$\left(x-1\right)\left(-x+1\right)\left(3x+2\right)$$