Find the partial fraction decomposition for the following.-
\[
\frac{-2 x-2}{x^{2}+2 x-3}=
\]
Question Help:
VIDEO
Written Exampise
Submit Question

Step 1 ：The denominator is a quadratic expression of the form \(ax^2 + bx + c\). We can factor it by finding two numbers that multiply to \(c\) (the constant term) and add to \(b\) (the coefficient of the linear term).

Step 2 ：The quadratic expression is \(x^2 + 2x - 3\). So, we need to find two numbers that multiply to -3 and add to 2. The numbers 3 and -1 satisfy these conditions. Therefore, the factored form of the denominator is \((x - 1)(x + 3)\).

Step 3 ：Now, we can set up the general form of the partial fraction decomposition and solve for \(A\) and \(B\).

Step 4 ：The roots of the denominator are 1 and -3. So, we set up the equation as \(A/(x - 1) + B/(x + 3) = (-2x - 2)/(x^2 + 2x - 3)\).

Step 5 ：Solving this equation, we get \(A = -1\) and \(B = -1\).

Step 6 ：Therefore, the partial fraction decomposition of the given rational function is \(-1/(x - 1) - 1/(x + 3)\).

Step 7 ：\(\boxed{\frac{-1}{x - 1} + \frac{-1}{x + 3}}\) is the final answer.