Consider the triangular-based prism on the front of the rectangular prism. This prism has five faces: a rectangle on the front, a rectangle on the left, a triangle on the bottom, a triangle on the top, and a rectangle on the back. The rectangle on the front measures $3 \times 12$ and so has area 36. The rectangle on the left measures $3 \times 5$ and so has area 15. The triangles on the top and bottom each are right-angled and have legs of length 5 and 12. This means that each has area $\frac{1}{2} \times 12 \times 5 = 30$. The rectangle on the back has height 3. The length of this rectangle is the length of the diagonal of the bottom face of the rectangular prism. By the Pythagorean Theorem, this length is $\sqrt{5^{2} + 12^{2}} = \sqrt{25 + 144} = \sqrt{169} = 13$. Thus, this rectangle is $3 \times 13$ and so has area 39. In total, the surface area of the triangular prism is thus $36 + 15 + 2 \times 30 + 39 = 150$.