Solve (4-t)*(sqrt{3}+frac{sqrt{3}}{2}t)*1/2

$$( 4 - t ) \times ( \sqrt { 3 } + \frac { \sqrt { 3 } } { 2 } t ) \times \frac { 1 } { 2 }$$

Evaluate

$$\frac{\sqrt{3}\left(4-t\right)\left(t+2\right)}{4}$$

Short Solution Steps

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

$$\left(4-t\right)\left(\sqrt{3}+\frac{\sqrt{3}t}{2}\right)\times \frac{1}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{3} times \frac{2}{2}.

$$\left(4-t\right)\left(\frac{2\sqrt{3}}{2}+\frac{\sqrt{3}t}{2}\right)\times \frac{1}{2}$$

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

$$\left(4-t\right)\times \frac{2\sqrt{3}+\sqrt{3}t}{2}\times \frac{1}{2}$$

Express \left(4-t\right)\times \frac{2\sqrt{3}+\sqrt{3}t}{2} as a single fraction.

$$\frac{\left(4-t\right)\left(2\sqrt{3}+\sqrt{3}t\right)}{2}\times \frac{1}{2}$$

Multiply \frac{\left(4-t\right)\left(2\sqrt{3}+\sqrt{3}t\right)}{2} times \frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(4-t\right)\left(2\sqrt{3}+\sqrt{3}t\right)}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{\left(4-t\right)\left(2\sqrt{3}+\sqrt{3}t\right)}{4}$$

Apply the distributive property by multiplying each term of 4-t by each term of 2\sqrt{3}+\sqrt{3}t.

$$\frac{8\sqrt{3}+4\sqrt{3}t-2\sqrt{3}t-\sqrt{3}t^{2}}{4}$$

Combine 4\sqrt{3}t and -2\sqrt{3}t to get 2\sqrt{3}t.

$$\frac{8\sqrt{3}+2\sqrt{3}t-\sqrt{3}t^{2}}{4}$$

Factor

$$\frac{\sqrt{3}\left(4-t\right)\left(t+2\right)}{4}$$