Solve a-2/a^2+2a-a-1/a^2+4a+4

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

Evaluate

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

Short Solution Steps

Factor a^{2}+2a. Factor a^{2}+4a+4.

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

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

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

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

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

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

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

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

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

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

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

Expand

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

Short Solution Steps

Factor a^{2}+2a. Factor a^{2}+4a+4.

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

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

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

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

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

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

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

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

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

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

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