Problem 73 - College Algebra

Question

Perform the indicated operations. Simplify the result, if possible. $$\left(\frac{2 x+3}{x+1} \cdot \frac{x^{2}+4 x-5}{2 x^{2}+x-3}\right)-\frac{2}{x+2}$$

Step-by-Step Solution

Answer

The final simplified answer is \( \frac{-3x^{3} + 5x^{2} + 13x -21}{(x+2)(2x^{3} + 3x^{2} - 3x -3)} \)

Step 1: Simplify the complex fraction

We first simplify and perform the multiplication in the complex fraction. This involves flipping the second fraction in the multiplication and changing it to a division sign, as follows: \( \frac{2 x+3}{x+1} \times \frac{x^{2}+4 x-5}{2 x^{2}+x-3} = \frac{(2x + 3) \times (x^{2}+4 x-5)}{ (x+1) \times (2x^{2} + x - 3)} \)

Step 2: Expand numerator and simplify

Expand the terms in the numerator of the new fraction to obtain \( 2x^{3} + 8x^{2} -5x + 3x^{2} + 12x -15 \) . And then simplify to get \( 2x^{3} + 11x^{2} + 7x -15 \)

Step 3: Expand denominator and simplify

Similarly, expand the terms in the denominator of the new fraction to get \(2x^{3} + 2x^{2} - 3x + x^{2} -3\) . Then simplify to obtain \( 2x^{3} + 3x^{2} - 3x -3 \)

Step 4: Subtraction of fractions

The original equation can now be written as \( \frac{2x^{3} + 11x^{2} + 7x -15}{2x^{3} + 3x^{2} - 3x -3} - \frac{2}{x+2} \) . For subtraction of fractions, we need the denominators to be the same. Multiply the second term by \(\frac{2x^{3} + 3x^{2} - 3x -3}{2x^{3} + 3x^{2} - 3x -3}\), yielding \(\frac{2(2x^{3} + 3x^{2} - 3x -3)}{(x+2)(2x^{3} + 3x^{2} - 3x -3)}\)

Step 5: Simplify

Now we can perform the subtraction to get, \( \frac{2x^{3} + 11x^{2} + 7x -15 - 2(2x^{3} + 3x^{2} - 3x -3)}{(x+2)(2x^{3} + 3x^{2} - 3x -3)} \). Simplify the above expression to get the final answer.