Simplify. Express your answer as a single fraction in simplest form. \[ 5 c+2-\frac{c}{6 c^{2}+33 c+1} \]

To simplify the expression \( 5c + 2 - \frac{c}{6c^2 + 33c + 1} \), we first need a common denominator. The first step is to express \( 5c + 2 \) as a fraction with the denominator \( 6c^2 + 33c + 1 \). We can do this by multiplying \( 5c + 2 \) by the common denominator: \[ 5c + 2 = \frac{(5c + 2)(6c^2 + 33c + 1)}{6c^2 + 33c + 1} \] Now, rewrite the whole expression: \[ \frac{(5c + 2)(6c^2 + 33c + 1)}{6c^2 + 33c + 1} - \frac{c}{6c^2 + 33c + 1} \] Since they share a common denominator, we can combine the numerators: \[ \frac{(5c + 2)(6c^2 + 33c + 1) - c}{6c^2 + 33c + 1} \] Now we need to simplify the numerator: 1. Expand \( (5c + 2)(6c^2 + 33c + 1) \): \[ = 5c \cdot 6c^2 + 5c \cdot 33c + 5c \cdot 1 + 2 \cdot 6c^2 + 2 \cdot 33c + 2 \cdot 1 \] \[ = 30c^3 + 165c^2 + 5c + 12c^2 + 66c + 2 \] \[ = 30c^3 + (165c^2 + 12c^2) + (5c + 66c) + 2 \] \[ = 30c^3 + 177c^2 + 71c + 2 \] 2. Now, subtract \( c \): \[ 30c^3 + 177c^2 + 71c + 2 - c = 30c^3 + 177c^2 + 70c + 2 \] Now, we write the combined expression: \[ \frac{30c^3 + 177c^2 + 70c + 2}{6c^2 + 33c + 1} \] To ensure the fraction is in its simplest form, we check if the numerator can be factored. While \( 6c^2 + 33c + 1 \) does not have simple factors, we can check for the greatest common divisor. After confirming, we find no common factors with the denominator. Thus, the simplified expression is: \[ \frac{30c^3 + 177c^2 + 70c + 2}{6c^2 + 33c + 1} \]