Simplify. Express your answer as a single fraction in simplest form \[ 4 d+\frac{2 d}{10 d^{2}+39 d+5} \]

To simplify the expression \( 4d + \frac{2d}{10d^2 + 39d + 5} \), we first need a common denominator. The expression can be rewritten as: \[ \frac{4d(10d^2 + 39d + 5)}{10d^2 + 39d + 5} + \frac{2d}{10d^2 + 39d + 5} \] Now, combine the fractions: \[ \frac{4d(10d^2 + 39d + 5) + 2d}{10d^2 + 39d + 5} \] Next, simplify the numerator: 1. Distribute \(4d\) in the first term: \[ 4d(10d^2 + 39d + 5) = 40d^3 + 156d^2 + 20d \] 2. Add \(2d\): \[ 40d^3 + 156d^2 + 20d + 2d = 40d^3 + 156d^2 + 22d \] Thus, the expression becomes: \[ \frac{40d^3 + 156d^2 + 22d}{10d^2 + 39d + 5} \] Next, we can factor the numerator. First, factor out \(2d\): \[ 40d^3 + 156d^2 + 22d = 2d(20d^2 + 78d + 11) \] Now, the expression is: \[ \frac{2d(20d^2 + 78d + 11)}{10d^2 + 39d + 5} \] At this point, we should see if \(20d^2 + 78d + 11\) and \(10d^2 + 39d + 5\) can be factored or simplified further. To simplify verified things, we can directly substitute numerical approaches, or use the discriminant of the quadratic formulas to test whether they have common roots. However, upon testing, it turns out that they cannot simplify gracefully as they do not share common roots. Thus, the final simplified expression is: \[ \frac{2d(20d^2 + 78d + 11)}{10d^2 + 39d + 5} \] This represents the given expression in simplest form as a single fraction.