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Chapter 0: Problem 83

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
\[ \frac {2x^2 +5x +3} {(x+2)(2x^3 +3x^2 -3x -3)} \]

Step by step solution

01

Perform Multiplication

Start by multiplying the first two fractions in the expression. Remember that multiplying fractions involves multiplying the numerators together and the denominators together. Thus, the multiplication will yield: \[ \frac {(2x+3)(x^2+4x-5)} {(x+1)(2x^2+x-3)} \] Keep the second fraction `-2/(x+2)` as it is for now.
02

Expression Simplification

Expanding both numerator and denominator by applying distributive property, we get: \[ \frac {2x^3 +11x^2 -4x -15} {2x^3 +3x^2 -3x -3}\] - \frac {2}{x+2}
03

Combine fractions

To subtract the second fraction from the first, you first need to find a common denominator. Since the denominators are different, multiply the second fraction by the denominator of the first one: \[ \frac {2(2x^3 +3x^2 -3x -3)} {(x+2)(2x^3 +3x^2 -3x -3)} \] Now perform the subtraction.
04

Perform Subtraction

After the denominators are equal, perform subtraction operation by subtracting the numerators, leading to: \[ \frac {(2x^3 +11x^2 -4x -15) -2(2x^3 +3x^2 -3x -3)} {(x+2)(2x^3 +3x^2 -3x -3)} \] Simplify this further.
05

Simplify Numerator

Simplify the numerator by applying the distributive property within the parentheses and then subtracting, gives us: \[ \frac {2x^2 +5x +3} {(x+2)(2x^3 +3x^2 -3x -3)} \] as the final simplified fraction.

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