Simplify. Express your answer as a single fraction in simplest form. \( 7 s+7+\frac{s}{s+10} \)

First, combine the terms \( 7s + 7 \) into a single fraction. To do this, we can rewrite \( 7s + 7 \) as \( \frac{(7s + 7)(s + 10)}{s + 10} \), which gives us a common denominator with the existing fraction. Now, distribute: \[ 7s + 7 = 7(s + 1) \] So, we express it as: \[ 7(s + 1) \cdot \frac{s + 10}{s + 10} = \frac{7(s + 1)(s + 10)}{s + 10} \] Now we can add both fractions together: \[ = \frac{7(s + 1)(s + 10) + s}{s + 10} \] Next, we will expand \( 7(s + 1)(s + 10) \): \[ = 7(s^2 + 11s + 10) + s = 7s^2 + 77s + 70 + s = 7s^2 + 78s + 70 \] Putting this all together, we have: \[ \frac{7s^2 + 78s + 70}{s + 10} \] Now, we need to check if the numerator can be factored. We can try to factor \( 7s^2 + 78s + 70 \). The potential factors of \( 7s^2 + 78s + 70 \) could be \( (7s + 10)(s + 7) \): \[ 7s^2 + 49s + 10s + 70 = 7s^2 + 78s + 70 \] So, we can express the fraction as: \[ \frac{(7s + 10)(s + 7)}{s + 10} \] Now, we can simplify this fraction as follows: \[ = (7s + 10)(s + 7) \frac{1}{(s + 10)} \] Thus, the simplified expression is: \[ \frac{(7s + 10)(s + 7)}{s + 10} \] This fraction is already in simplest form given that \( s + 10 \) cannot cancel with any other terms. So the final answer is: \(\frac{(7s + 10)(s + 7)}{s + 10}\)