Solve x-3/x^2-4-x+1/x^2+4x+4+4/(x+2)(x^2-4)

$$\frac { x - 3 } { x ^ { 2 } - 4 } - \frac { x + 1 } { x ^ { 2 } + 4 x + 4 } + \frac { 4 } { ( x + 2 ) ( x ^ { 2 } - 4 ) }$$

Evaluate

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Short Solution Steps

Factor x^{2}-4. Factor x^{2}+4x+4.

$$\frac{x-3}{\left(x-2\right)\left(x+2\right)}-\frac{x+1}{\left(x+2\right)^{2}}+\frac{4}{\left(x+2\right)\left(x^{2}-4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+2\right) and \left(x+2\right)^{2} is \left(x-2\right)\left(x+2\right)^{2}. Multiply \frac{x-3}{\left(x-2\right)\left(x+2\right)} times \frac{x+2}{x+2}. Multiply \frac{x+1}{\left(x+2\right)^{2}} times \frac{x-2}{x-2}.

$$\frac{\left(x-3\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)^{2}}-\frac{\left(x+1\right)\left(x-2\right)}{\left(x-2\right)\left(x+2\right)^{2}}+\frac{4}{\left(x+2\right)\left(x^{2}-4\right)}$$

Since \frac{\left(x-3\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)^{2}} and \frac{\left(x+1\right)\left(x-2\right)}{\left(x-2\right)\left(x+2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x-3\right)\left(x+2\right)-\left(x+1\right)\left(x-2\right)}{\left(x-2\right)\left(x+2\right)^{2}}+\frac{4}{\left(x+2\right)\left(x^{2}-4\right)}$$

Do the multiplications in \left(x-3\right)\left(x+2\right)-\left(x+1\right)\left(x-2\right).

$$\frac{x^{2}+2x-3x-6-x^{2}+2x-x+2}{\left(x-2\right)\left(x+2\right)^{2}}+\frac{4}{\left(x+2\right)\left(x^{2}-4\right)}$$

Combine like terms in x^{2}+2x-3x-6-x^{2}+2x-x+2.

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

Factor \left(x+2\right)\left(x^{2}-4\right).

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

Since \frac{-4}{\left(x-2\right)\left(x+2\right)^{2}} and \frac{4}{\left(x-2\right)\left(x+2\right)^{2}} have the same denominator, add them by adding their numerators. Add -4 and 4 to get 0.

$$\frac{0}{\left(x-2\right)\left(x+2\right)^{2}}$$

Zero divided by any non-zero term gives zero.

$$0$$

Factor

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