Solve -frac{sqrt{3}}{3}m^2-frac{2sqrt{3}}{3}m+frac{sqrt{3}}{3}m+frac{2sqrt{3}}{3}

$$- \frac { \sqrt { 3 } } { 3 } m ^ { 2 } - \frac { 2 \sqrt { 3 } } { 3 } m + \frac { \sqrt { 3 } } { 3 } m + \frac { 2 \sqrt { 3 } } { 3 }$$

Evaluate

$$\frac{\sqrt{3}\left(1-m\right)\left(m+2\right)}{3}$$

Short Solution Steps

Express \left(-\frac{\sqrt{3}}{3}\right)m^{2} as a single fraction.

$$\frac{-\sqrt{3}m^{2}}{3}-\frac{2\sqrt{3}}{3}m+\frac{\sqrt{3}}{3}m+\frac{2\sqrt{3}}{3}$$

Express \frac{2\sqrt{3}}{3}m as a single fraction.

$$\frac{-\sqrt{3}m^{2}}{3}-\frac{2\sqrt{3}m}{3}+\frac{\sqrt{3}}{3}m+\frac{2\sqrt{3}}{3}$$

Since \frac{-\sqrt{3}m^{2}}{3} and \frac{2\sqrt{3}m}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\sqrt{3}m^{2}-2\sqrt{3}m}{3}+\frac{\sqrt{3}}{3}m+\frac{2\sqrt{3}}{3}$$

Express \frac{\sqrt{3}}{3}m as a single fraction.

$$\frac{-\sqrt{3}m^{2}-2\sqrt{3}m}{3}+\frac{\sqrt{3}m}{3}+\frac{2\sqrt{3}}{3}$$

Since \frac{-\sqrt{3}m^{2}-2\sqrt{3}m}{3} and \frac{\sqrt{3}m}{3} have the same denominator, add them by adding their numerators.

$$\frac{-\sqrt{3}m^{2}-2\sqrt{3}m+\sqrt{3}m}{3}+\frac{2\sqrt{3}}{3}$$

Combine like terms in -\sqrt{3}m^{2}-2\sqrt{3}m+\sqrt{3}m.

$$\frac{-\sqrt{3}m^{2}-\sqrt{3}m}{3}+\frac{2\sqrt{3}}{3}$$

Since \frac{-\sqrt{3}m^{2}-\sqrt{3}m}{3} and \frac{2\sqrt{3}}{3} have the same denominator, add them by adding their numerators.

$$\frac{-\sqrt{3}m^{2}-\sqrt{3}m+2\sqrt{3}}{3}$$

Factor

$$\frac{\sqrt{3}\left(1-m\right)\left(m+2\right)}{3}$$

Solution Steps

Factor out \frac{1}{3}.

$$\frac{-\sqrt{3}m^{2}-2\sqrt{3}m+\sqrt{3}m+2\sqrt{3}}{3}$$

Consider -\sqrt{3}m^{2}-2\sqrt{3}m+\sqrt{3}m+2\sqrt{3}. Factor out \sqrt{3}.

$$\sqrt{3}\left(-m^{2}-2m+m+2\right)$$

Consider -m^{2}-2m+m+2. Multiply and combine like terms.

$$-m^{2}-m+2$$

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

$$a+b=-1 ab=-2=-2$$

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. The only such pair is the system solution.

$$a=1 b=-2$$

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

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

Factor out m in the first and 2 in the second group.

$$m\left(-m+1\right)+2\left(-m+1\right)$$

Factor out common term -m+1 by using distributive property.

$$\left(-m+1\right)\left(m+2\right)$$

Rewrite the complete factored expression.

$$\frac{\sqrt{3}\left(-m+1\right)\left(m+2\right)}{3}$$