Subtract x/(x^2-16)-8/(x^2+5x+4)

Problem

x/(x2−16)−8/(x2+5*x+4)

Solution

1. Factor the denominators of both fractions to identify the least common denominator.

x2−16=(x−4)*(x+4)

x2+5*x+4=(x+1)*(x+4)

2. Identify the least common denominator (LCD) by taking the product of all unique factors.

LCD=(x−4)*(x+4)*(x+1)

3. Rewrite each fraction with the LCD by multiplying the numerator and denominator by the missing factors.

(x*(x+1))/((x−4)*(x+4)*(x+1))−(8*(x−4))/((x+1)*(x+4)*(x−4))

4. Distribute the terms in the numerators.

(x2+x)/((x−4)*(x+4)*(x+1))−(8*x−32)/((x−4)*(x+4)*(x+1))

5. Subtract the numerators and combine them over the common denominator.

(x2+x−(8*x−32))/((x−4)*(x+4)*(x+1))

6. Simplify the numerator by combining like terms.

(x2−7*x+32)/((x−4)*(x+4)*(x+1))

Final Answer

x/(x2−16)−8/(x2+5*x+4)=(x2−7*x+32)/((x−4)*(x+4)*(x+1))

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