Poisson–Gamma posterior predictive is Negative Binomial

Suppose we have 10 observations $\underline y=(y_1,\dots,y_{10})$ from Poisson$(\lambda)$ and we assume a Gamma$(a,b)$ prior for $\lambda$. Then the posterior predictive distribution $p(y_{new}/\underline y)$ of a new observation $y_{new}$ is given by: