Problem 57 - College Algebra

Question

Add or subtract as indicated. $$\frac{4 x^{2}+x-6}{x^{2}+3 x+2}-\frac{3 x}{x+1}+\frac{5}{x+2}$$

Step-by-Step Solution

Answer

The fraction simplified is \(\frac{{x^2 - x + 1}}{{(x+1)(x+2)}}\).

Step 1: Factorize the Denominators

Factorize the denominators to get: \[\frac{{4x^2 + x - 6}}{{(x+2)(x+1)}} - \frac{3x}{{x+1}} + \frac{5}{{x+2}}\]

Step 2: Find Common Denominator

In order to add or subtract fractions the denominators must be the same. The common denominator for the fractions here is \((x+1)(x+2)\)

Step 3: Rewrite with Common Denominator

Transform each fraction so they all have the common denominator:\[\frac{{4x^2 + x - 6}}{{(x+1)(x+2)}} - \frac{3x(x+2)}{{(x+1)(x+2)}} + \frac{5(x+1)}{{(x+1)(x+2)}\]

Step 4: Combine the Fractions

Now, combine the fractions and simplify the numerator:\[\frac{{4x^2 + x - 6 - 3x^2 - 6x + 5x + 5}}{{(x+1)(x+2)}}\]which simplifies to:\[\frac{{x^2 - x + 1}}{{(x+1)(x+2)}}\]