Problem 73 - College Algebra Essentials

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

\[ \frac{{2x^3 + 7x^2 + 2x - 6}}{(2x - 1)(x + 3)(x + 2)} \]

Step 1 Simplify the Multiplication

The first operation to be performed is multiplication of the two rational expressions. So firstly, perform the multiplication operation to the two fractions to form a single fraction.\[ \frac{{(2x+3)(x^2 + 4x - 5)}}{{(x+1)(2x^2 + x -3)}} \]

Step 2 Factorize the Numerator and Denominator

Next step is to factorize both the numerator and the denominator of the fraction.\[ \frac{{(2x + 3)(x - 1)(x + 5)}}{{( x - 1)(2x - 1)(x + 3)}} \]

Step 3 Simplify the Fraction

Now, cancel out mutual terms from numerator and denominator.\[ \frac{{(2x + 3)(x + 5)}}{{(2x - 1)(x + 3)}} \]

Step 4 Perform the Subtraction operation

The fraction now has to subtract the rational expression \( \frac{2}{x+2} \) from it.\[ \frac{{(2x + 3)(x + 5)}}{{(2x - 1)(x + 3)}} - \frac{2}{x+2} \]

Step 5 Make a Common Denominator

After having subtracted two rational expressions, the next step is to have a common denominator.\[ \frac{{(2x + 3)(x + 5)(x+2) - 2(2x - 1)(x + 3)}}{{(2x - 1)(x + 3)(x+2)}} \]

Step 6 Simplify the expression

Lastly, simplify the above expression to get the final answer.\[ \frac{{2x^3 + 7x^2 + 2x - 6}}{(2x - 1)(x + 3)(x + 2)} \]