Simplify. Express your answer as a single fraction in simplest form. \( \frac{u}{8 u+1}-\frac{3 u}{u+1} \)

To simplify the expression \( \frac{u}{8u+1} - \frac{3u}{u+1} \), we first need to find a common denominator. The denominators are \( 8u + 1 \) and \( u + 1 \). The common denominator will be \( (8u + 1)(u + 1) \). Rewriting each fraction with the common denominator gives us: \[ \frac{u(u + 1)}{(8u + 1)(u + 1)} - \frac{3u(8u + 1)}{(8u + 1)(u + 1)} \] Now, we can combine the numerators: \[ \frac{u(u + 1) - 3u(8u + 1)}{(8u + 1)(u + 1)} \] Expanding the numerators: 1. For \( u(u + 1) \): \[ u^2 + u \] 2. For \( -3u(8u + 1) \): \[ -24u^2 - 3u \] So, the combined numerator is: \[ u^2 + u - 24u^2 - 3u = (1 - 24)u^2 + (1 - 3)u = -23u^2 - 2u \] Thus, the expression is: \[ \frac{-23u^2 - 2u}{(8u + 1)(u + 1)} \] Now, we can factor out a \(-1\) from the numerator: \[ \frac{-(23u^2 + 2u)}{(8u + 1)(u + 1)} \] Factoring further, we have: \[ 23u^2 + 2u = u(23u + 2) \] So our expression now looks like: \[ \frac{-u(23u + 2)}{(8u + 1)(u + 1)} \] Now, we write the final answer: \[ \frac{-u(23u + 2)}{(8u + 1)(u + 1)} \] This is the simplified expression in simplest form.