Let P(x) be a polynomial whose degree is 3. If P(1) = 1, P(2) = 3, P(3) = 2, and P(4) = 10, compute the value P(5).

This is similar  to this  problem I just worked  :

https://web2.0calc.com/questions/if-f-is-a-polynomial-of-degree-3-such-that-f-0-f_1

We  have this  system

a  +  b  +  c +  d   =  1

8a  + 4b + 2c  + d  = 3

27a + 9b + 3c + d  = 2

64a +16b + 4c  + d  =10

Time doesn't permit me  to  go  through the solving process (it isn't difficult, just tedious) but WoframAlpha  gives  the solutions:

a  =2,  b = -27/2  c   =57/2 d  = -16

Our polynomial is   f(x) = 2x^3  - (27/2)x^2 + (57/2)x   - 16

P(5)   =  39

See the graph  here : https://www.desmos.com/calculator/6t4hhafji0