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Chapter 0: Problem 73

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

Short Answer

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

Step by step solution

01

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: \( \frac{2 x+3}{x+1} \times \frac{x^{2}+4 x-5}{2 x^{2}+x-3} = \frac{(2x + 3) \times (x^{2}+4 x-5)}{ (x+1) \times (2x^{2} + x - 3)} \)
02

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 \( 2x^{3} + 11x^{2} + 7x -15 \)
03

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 \( 2x^{3} + 3x^{2} - 3x -3 \)
04

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 \(\frac{2(2x^{3} + 3x^{2} - 3x -3)}{(x+2)(2x^{3} + 3x^{2} - 3x -3)}\)
05

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.

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