<s>[INST] $P(x) = x^4 + ax^3 +bx^2 +cx+d$ Let $P(x) = x^4 + ax^3 +bx^2 +cx+d$ where $a, b, c, d$ are integers.
$P(x)$ is divided by $x-2012, x-2013, x-2014, x-2015, x-2016$ and has the remainders $24, -6, 4, -6, 24$ respectively. 
What is the remainder when $P(x)$ is divided by $x-2017$ ?
Is my answer correct ?
$P(2012) = 24, \;P(2013) = -6, \;P(2014) = 4, \;P(2015) = -6, \;P(2016) = 24$
By Lagrange interpolation,
$P(2017) = \sum{24\cdot\frac{(2017-2013)(2017-2014)(2017-2015)(2017-2016)}{(2012-2013)(2012-2014)(2012-2015)(2012-2016)}} = 24+30+40+60+120=274$
 [/INST] By symmetry at $x=2014$ we can write $\ P(x) = 4 + a(x\!-\!2014)^2 + b(x\!-\!2014)^4$. 
Evaluation implies that $\ a=-15,\ b=5\ $ so $\,P(2017)=4-15(9)+5(81)=274$
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