Solve (a^2/2-a+1)*(a+1+a/2

$$( \frac { a ^ { 2 } } { 2 } - a + 1 ) \cdot ( a + 1 + \frac { a } { 2 }$$

Evaluate

$$\frac{3a^{3}}{4}-a^{2}+\frac{a}{2}+1$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply -a+1 times \frac{2}{2}.

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

Since \frac{a^{2}}{2} and \frac{2\left(-a+1\right)}{2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in a^{2}+2\left(-a+1\right).

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

Combine a and \frac{a}{2} to get \frac{3}{2}a.

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

Use the distributive property to multiply \frac{a^{2}-2a+2}{2} by \frac{3}{2}a+1.

$$\frac{3}{2}\times \frac{a^{2}-2a+2}{2}a+\frac{a^{2}-2a+2}{2}$$

Multiply \frac{3}{2} times \frac{a^{2}-2a+2}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\left(a^{2}-2a+2\right)}{2\times 2}a+\frac{a^{2}-2a+2}{2}$$

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

$$\frac{3\left(a^{2}-2a+2\right)a}{2\times 2}+\frac{a^{2}-2a+2}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\times 2 and 2 is 2\times 2. Multiply \frac{a^{2}-2a+2}{2} times \frac{2}{2}.

$$\frac{3\left(a^{2}-2a+2\right)a}{2\times 2}+\frac{2\left(a^{2}-2a+2\right)}{2\times 2}$$

Since \frac{3\left(a^{2}-2a+2\right)a}{2\times 2} and \frac{2\left(a^{2}-2a+2\right)}{2\times 2} have the same denominator, add them by adding their numerators.

$$\frac{3\left(a^{2}-2a+2\right)a+2\left(a^{2}-2a+2\right)}{2\times 2}$$

Do the multiplications in 3\left(a^{2}-2a+2\right)a+2\left(a^{2}-2a+2\right).

$$\frac{3a^{3}-6a^{2}+6a+2a^{2}-4a+4}{2\times 2}$$

Combine like terms in 3a^{3}-6a^{2}+6a+2a^{2}-4a+4.

$$\frac{3a^{3}-4a^{2}+2a+4}{2\times 2}$$

Expand 2\times 2.

$$\frac{3a^{3}-4a^{2}+2a+4}{4}$$

Expand

$$\frac{3a^{3}}{4}-a^{2}+\frac{a}{2}+1$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply -a+1 times \frac{2}{2}.

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

Since \frac{a^{2}}{2} and \frac{2\left(-a+1\right)}{2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in a^{2}+2\left(-a+1\right).

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

Combine a and \frac{a}{2} to get \frac{3}{2}a.

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

Use the distributive property to multiply \frac{a^{2}-2a+2}{2} by \frac{3}{2}a+1.

$$\frac{3}{2}\times \frac{a^{2}-2a+2}{2}a+\frac{a^{2}-2a+2}{2}$$

Multiply \frac{3}{2} times \frac{a^{2}-2a+2}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\left(a^{2}-2a+2\right)}{2\times 2}a+\frac{a^{2}-2a+2}{2}$$

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

$$\frac{3\left(a^{2}-2a+2\right)a}{2\times 2}+\frac{a^{2}-2a+2}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\times 2 and 2 is 2\times 2. Multiply \frac{a^{2}-2a+2}{2} times \frac{2}{2}.

$$\frac{3\left(a^{2}-2a+2\right)a}{2\times 2}+\frac{2\left(a^{2}-2a+2\right)}{2\times 2}$$

Since \frac{3\left(a^{2}-2a+2\right)a}{2\times 2} and \frac{2\left(a^{2}-2a+2\right)}{2\times 2} have the same denominator, add them by adding their numerators.

$$\frac{3\left(a^{2}-2a+2\right)a+2\left(a^{2}-2a+2\right)}{2\times 2}$$

Do the multiplications in 3\left(a^{2}-2a+2\right)a+2\left(a^{2}-2a+2\right).

$$\frac{3a^{3}-6a^{2}+6a+2a^{2}-4a+4}{2\times 2}$$

Combine like terms in 3a^{3}-6a^{2}+6a+2a^{2}-4a+4.

$$\frac{3a^{3}-4a^{2}+2a+4}{2\times 2}$$

Expand 2\times 2.

$$\frac{3a^{3}-4a^{2}+2a+4}{4}$$