Simplify. Express your answer as a single fraction in simplest form \( 5-\frac{c}{c^{2}-34 c-32} \)

To simplify the expression \( 5 - \frac{c}{c^{2}-34 c-32} \), we first need a common denominator. We start by factoring the denominator \( c^2 - 34c - 32 \). To factor \( c^2 - 34c - 32 \), we look for two numbers that multiply to \(-32\) and add to \(-34\). These numbers are \(-36\) and \(2\). Therefore, we can write: \[ c^2 - 34c - 32 = (c - 36)(c + 2) \] Now, we rewrite \( 5 \) with the common denominator: \[ 5 = \frac{5(c - 36)(c + 2)}{(c - 36)(c + 2)} \] Now, substituting this back into the original expression: \[ 5 - \frac{c}{(c - 36)(c + 2)} = \frac{5(c - 36)(c + 2) - c}{(c - 36)(c + 2)} \] Next, we simplify the numerator: \[ 5(c - 36)(c + 2) - c = 5(c^2 + 2c - 36c - 72) - c = 5(c^2 - 34c - 72) - c \] Now, distributing \( 5 \): \[ 5c^2 - 170c - 360 - c = 5c^2 - 171c - 360 \] So the expression becomes: \[ \frac{5c^2 - 171c - 360}{(c - 36)(c + 2)} \] Now, we can’t simplify \( 5c^2 - 171c - 360 \) further without factorization, but we can check if it has integer factors using the quadratic formula or factor methods if needed. Thus, the final simplified expression is: \[ \frac{5c^2 - 171c - 360}{(c - 36)(c + 2)} \] This is expressed as a single fraction in simplest form, assuming further factorization of the quadratic doesn't yield simpler terms.