Question: Add or subtract as indicated. \[ \frac{5x}{x + 2} + \frac{17}{x - 2} - \frac{40}{x^2 - 4} \]…

Add or subtract as indicated. \[ \frac{5x}{x + 2} + \frac{17}{x - 2} - \frac{40}{x^2 - 4} \] \[ \frac{5x}{x + 2} + \frac{17}{x - 2} - \frac{40}{x^2 - 4} = \boxed{\text{(Simplify your answer.)}} \]

Solution

To solve the expression \( \frac{5x}{x+2} + \frac{17}{x-2} - \frac{40}{x^2 - 4} \), we need to perform algebraic operations and simplify.

First, recognize that \( x^2 - 4 \) can be factored as \((x+2)(x-2)\).

Rewrite the expression using this factorization:

\[ \frac{5x}{x+2} + \frac{17}{x-2} - \frac{40}{(x+2)(x-2)} \]

Find the common denominator, which is \((x+2)(x-2)\).

Rewrite each fraction with the common denominator:

\[ \frac{5x(x-2)}{(x+2)(x-2)} + \frac{17(x+2)}{(x+2)(x-2)} - \frac{40}{(x+2)(x-2)} \]

Now combine the fractions:

\[ \frac{5x(x-2) + 17(x+2) - 40}{(x+2)(x-2)} \]

Distribute and simplify the numerator:

\[ 5x(x-2) = 5x^2 - 10x \]

\[ 17(x+2) = 17x + 34 \]

Combine these in the numerator:

\[ 5x^2 - 10x + 17x + 34 - 40 \]

Simplify by combining like terms:

\[ 5x^2 + 7x - 6 \]

Now place the simplified numerator over the denominator:

\[ \frac{5x^2 + 7x - 6}{(x+2)(x-2)} \]

This is the simplified form of the expression.