A rectangle with positive integer side lengths in cm has area $A$ cm$^2$ and perimeter $P$ cm. Which of $150,152,154,156,158$ cannot equal $A+P$?

1. $A+P=(a+2)(b+2)-4=N$ is achievable iff $N+4$ has a factorization into two factors both $\ge3$.
2. $158+4=162=2\cdot3^4$: factor pairs with both $\ge3$ include $(3,54),(6,27),(9,18)$ -- achievable. $156+4=160=2^5\cdot5$: $(4,40),(5,32),(8,20),(10,16)$ -- achievable.
3. $154+4=158=2\cdot79$ with $79$ prime: only factor pairs are $(1,158),(2,79)$, neither with both factors $\ge3$ -- \emph{not} achievable.
4. $152+4=156=2^2\cdot3\cdot13$: $(3,52),(4,39),(6,26),(12,13)$ -- achievable. $150+4=154=2\cdot7\cdot11$: $(7,22),(11,14)$ -- achievable.
5. So $154$ is the one that cannot equal $A+P$.