Problem

simplify and state restriction
\[
\frac{x}{x^{2}-3 x-4}-\frac{4}{x+1}
\]

Answer

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Answer

So, the simplified form of the given expression is \[\boxed{\frac{-3x+16}{(x-4)(x+1)}}\] with restrictions \(x\neq4\) and \(x\neq-1\).

Steps

Step 1 :Factor the quadratic in the denominator of the first fraction. The quadratic \(x^{2}-3x-4\) can be factored into \((x-4)(x+1)\). So, the expression becomes: \[\frac{x}{(x-4)(x+1)}-\frac{4}{x+1}\]

Step 2 :Find a common denominator for the two fractions. In this case, the common denominator is \((x-4)(x+1)\). So, we rewrite the second fraction with this common denominator: \[\frac{x}{(x-4)(x+1)}-\frac{4(x-4)}{(x-4)(x+1)}\]

Step 3 :Combine the numerators since the fractions have the same denominator: \[\frac{x-4x+16}{(x-4)(x+1)}\]

Step 4 :Simplify the numerator to get: \[\frac{-3x+16}{(x-4)(x+1)}\]

Step 5 :The restrictions are the values of \(x\) that make the denominator equal to zero, because division by zero is undefined. Setting each factor in the denominator equal to zero gives: \(x-4=0\) --> \(x=4\) and \(x+1=0\) --> \(x=-1\). So, the restrictions are \(x\neq4\) and \(x\neq-1\).

Step 6 :So, the simplified form of the given expression is \[\boxed{\frac{-3x+16}{(x-4)(x+1)}}\] with restrictions \(x\neq4\) and \(x\neq-1\).

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