Solve a^2/a^2+3a+2-2a/a+2

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

Evaluate

$$-\frac{a}{a+1}$$

Short Solution Steps

Factor a^{2}+3a+2.

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

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

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

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

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

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

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

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

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

Factor the expressions that are not already factored in \frac{-a^{2}-2a}{\left(a+1\right)\left(a+2\right)}.

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

Extract the negative sign in -2-a.

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

Cancel out a+2 in both numerator and denominator.

$$\frac{-a}{a+1}$$

Factor

$$-\frac{a}{a+1}$$