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

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

Evaluate

$$\frac{a^{3}-a^{2}-a-8}{3a\left(a+2\right)^{2}}$$

Short Solution Steps

Add 1 and 2 to get 3.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\times 3 and \left(a+2\right)^{2} is 3a\left(a+2\right)^{2}. Multiply \frac{a-2}{a\times 3} times \frac{\left(a+2\right)^{2}}{\left(a+2\right)^{2}}. Multiply \frac{a-1}{\left(a+2\right)^{2}} times \frac{3a}{3a}.

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

Since \frac{\left(a-2\right)\left(a+2\right)^{2}}{3a\left(a+2\right)^{2}} and \frac{\left(a-1\right)\times 3a}{3a\left(a+2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(a-2\right)\left(a+2\right)^{2}-\left(a-1\right)\times 3a.

$$\frac{a^{3}+4a^{2}+4a-2a^{2}-8a-8-3a^{2}+3a}{3a\left(a+2\right)^{2}}$$

Combine like terms in a^{3}+4a^{2}+4a-2a^{2}-8a-8-3a^{2}+3a.

$$\frac{a^{3}-a^{2}-a-8}{3a\left(a+2\right)^{2}}$$

Expand 3a\left(a+2\right)^{2}.

$$\frac{a^{3}-a^{2}-a-8}{3a^{3}+12a^{2}+12a}$$

Expand

$$\frac{a^{3}-a^{2}-a-8}{3a\left(a+2\right)^{2}}$$

Short Solution Steps

Add 1 and 2 to get 3.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\times 3 and \left(a+2\right)^{2} is 3a\left(a+2\right)^{2}. Multiply \frac{a-2}{a\times 3} times \frac{\left(a+2\right)^{2}}{\left(a+2\right)^{2}}. Multiply \frac{a-1}{\left(a+2\right)^{2}} times \frac{3a}{3a}.

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

Since \frac{\left(a-2\right)\left(a+2\right)^{2}}{3a\left(a+2\right)^{2}} and \frac{\left(a-1\right)\times 3a}{3a\left(a+2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(a-2\right)\left(a+2\right)^{2}-\left(a-1\right)\times 3a.

$$\frac{a^{3}+4a^{2}+4a-2a^{2}-8a-8-3a^{2}+3a}{3a\left(a+2\right)^{2}}$$

Combine like terms in a^{3}+4a^{2}+4a-2a^{2}-8a-8-3a^{2}+3a.

$$\frac{a^{3}-a^{2}-a-8}{3a\left(a+2\right)^{2}}$$

Expand 3a\left(a+2\right)^{2}.

$$\frac{a^{3}-a^{2}-a-8}{3a^{3}+12a^{2}+12a}$$