Solve =a/a+1-a-lambda/adiv(a/a+2-1/a^2+2a)

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

Evaluate

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

Short Solution Steps

Factor a^{2}+2a.

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

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

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

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

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

Do the multiplications in aa-1.

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

Divide \frac{a-\lambda }{a} by \frac{a^{2}-1}{a\left(a+2\right)} by multiplying \frac{a-\lambda }{a} by the reciprocal of \frac{a^{2}-1}{a\left(a+2\right)}.

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

Cancel out a in both numerator and denominator.

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

Factor a^{2}-1.

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

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

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

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

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

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

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

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

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

Expand \left(a-1\right)\left(a+1\right).

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

Expand

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

Short Solution Steps

Factor a^{2}+2a.

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

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

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

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

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

Do the multiplications in aa-1.

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

Divide \frac{a-\lambda }{a} by \frac{a^{2}-1}{a\left(a+2\right)} by multiplying \frac{a-\lambda }{a} by the reciprocal of \frac{a^{2}-1}{a\left(a+2\right)}.

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

Cancel out a in both numerator and denominator.

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

Factor a^{2}-1.

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

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

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

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

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

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

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

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

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

Expand \left(a-1\right)\left(a+1\right).

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