Solve frac{3a^{2}-7a+2}{a^{2}-a-2}*a^2+2a+1/3a^2+2a-1div9a^2-1/9a^2-3a

$$\frac { 3 a ^ { 2 } - 7 a + 2 } { a ^ { 2 } - a - 2 } \times \frac { a ^ { 2 } + 2 a + 1 } { 3 a ^ { 2 } + 2 a - 1 } \div \frac { 9 a ^ { 2 } - 1 } { 9 a ^ { 2 } - 3 a }$$

Evaluate

$$\frac{3a}{3a+1}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{3a^{2}-7a+2}{a^{2}-a-2}.

$$\frac{\frac{\left(a-2\right)\left(3a-1\right)}{\left(a-2\right)\left(a+1\right)}\times \frac{a^{2}+2a+1}{3a^{2}+2a-1}}{\frac{9a^{2}-1}{9a^{2}-3a}}$$

Cancel out a-2 in both numerator and denominator.

$$\frac{\frac{3a-1}{a+1}\times \frac{a^{2}+2a+1}{3a^{2}+2a-1}}{\frac{9a^{2}-1}{9a^{2}-3a}}$$

Factor the expressions that are not already factored in \frac{a^{2}+2a+1}{3a^{2}+2a-1}.

$$\frac{\frac{3a-1}{a+1}\times \frac{\left(a+1\right)^{2}}{\left(3a-1\right)\left(a+1\right)}}{\frac{9a^{2}-1}{9a^{2}-3a}}$$

Cancel out a+1 in both numerator and denominator.

$$\frac{\frac{3a-1}{a+1}\times \frac{a+1}{3a-1}}{\frac{9a^{2}-1}{9a^{2}-3a}}$$

Multiply \frac{3a-1}{a+1} times \frac{a+1}{3a-1} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(3a-1\right)\left(a+1\right)}{\left(a+1\right)\left(3a-1\right)}}{\frac{9a^{2}-1}{9a^{2}-3a}}$$

Cancel out \left(3a-1\right)\left(a+1\right) in both numerator and denominator.

$$\frac{1}{\frac{9a^{2}-1}{9a^{2}-3a}}$$

Factor the expressions that are not already factored in \frac{9a^{2}-1}{9a^{2}-3a}.

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

Cancel out 3a-1 in both numerator and denominator.

$$\frac{1}{\frac{3a+1}{3a}}$$

Divide 1 by \frac{3a+1}{3a} by multiplying 1 by the reciprocal of \frac{3a+1}{3a}.

$$\frac{3a}{3a+1}$$

Expand

$$\frac{3a}{3a+1}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{3a^{2}-7a+2}{a^{2}-a-2}.

$$\frac{\frac{\left(a-2\right)\left(3a-1\right)}{\left(a-2\right)\left(a+1\right)}\times \frac{a^{2}+2a+1}{3a^{2}+2a-1}}{\frac{9a^{2}-1}{9a^{2}-3a}}$$

Cancel out a-2 in both numerator and denominator.

$$\frac{\frac{3a-1}{a+1}\times \frac{a^{2}+2a+1}{3a^{2}+2a-1}}{\frac{9a^{2}-1}{9a^{2}-3a}}$$

Factor the expressions that are not already factored in \frac{a^{2}+2a+1}{3a^{2}+2a-1}.

$$\frac{\frac{3a-1}{a+1}\times \frac{\left(a+1\right)^{2}}{\left(3a-1\right)\left(a+1\right)}}{\frac{9a^{2}-1}{9a^{2}-3a}}$$

Cancel out a+1 in both numerator and denominator.

$$\frac{\frac{3a-1}{a+1}\times \frac{a+1}{3a-1}}{\frac{9a^{2}-1}{9a^{2}-3a}}$$

Multiply \frac{3a-1}{a+1} times \frac{a+1}{3a-1} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(3a-1\right)\left(a+1\right)}{\left(a+1\right)\left(3a-1\right)}}{\frac{9a^{2}-1}{9a^{2}-3a}}$$

Cancel out \left(3a-1\right)\left(a+1\right) in both numerator and denominator.

$$\frac{1}{\frac{9a^{2}-1}{9a^{2}-3a}}$$

Factor the expressions that are not already factored in \frac{9a^{2}-1}{9a^{2}-3a}.

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

Cancel out 3a-1 in both numerator and denominator.

$$\frac{1}{\frac{3a+1}{3a}}$$

Divide 1 by \frac{3a+1}{3a} by multiplying 1 by the reciprocal of \frac{3a+1}{3a}.

$$\frac{3a}{3a+1}$$