Solve (a-3a+4/a+3)diva-2/a+3-a+3/a+2

$$( a - \frac { 3 a + 4 } { a + 3 } ) \div \frac { a - 2 } { a + 3 } - \frac { a + 3 } { a + 2 }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply a times \frac{a+3}{a+3}.

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

Since \frac{a\left(a+3\right)}{a+3} and \frac{3a+4}{a+3} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in a\left(a+3\right)-\left(3a+4\right).

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

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

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

Divide \frac{a^{2}-4}{a+3} by \frac{a-2}{a+3} by multiplying \frac{a^{2}-4}{a+3} by the reciprocal of \frac{a-2}{a+3}.

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

Cancel out a+3 in both numerator and denominator.

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

Factor the expressions that are not already factored in \frac{a^{2}-4}{a-2}.

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

Cancel out a-2 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply a+2 times \frac{a+2}{a+2}.

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

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

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

Do the multiplications in \left(a+2\right)\left(a+2\right)-\left(a+3\right).

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

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

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply a times \frac{a+3}{a+3}.

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

Since \frac{a\left(a+3\right)}{a+3} and \frac{3a+4}{a+3} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in a\left(a+3\right)-\left(3a+4\right).

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

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

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

Divide \frac{a^{2}-4}{a+3} by \frac{a-2}{a+3} by multiplying \frac{a^{2}-4}{a+3} by the reciprocal of \frac{a-2}{a+3}.

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

Cancel out a+3 in both numerator and denominator.

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

Factor the expressions that are not already factored in \frac{a^{2}-4}{a-2}.

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

Cancel out a-2 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply a+2 times \frac{a+2}{a+2}.

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

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

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

Do the multiplications in \left(a+2\right)\left(a+2\right)-\left(a+3\right).

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

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

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