Determine the Cholesky factorization (without pivoting) of the matrix

$$A=\left[\begin{array}{ccc} 2 & -4 & 2 \\ -4 & 10+\lambda & 2+3 \lambda \\ 2 & 2+3 \lambda & 23+9 \lambda \end{array}\right]$$

where $\lambda$ is a real parameter. Hence, find the range of values of $\lambda$ for which the matrix $A$ is positive definite.