To find the remainder when polynomial P(x) = x⁵¹ + 51 is divided by (x + 1), we use the Remainder Theorem. This theorem states that the remainder of P(x) divided by (x - a) is P(a). Here, the divisor is (x + 1), which is (x - (-1)), so a = -1.

Therefore, we compute P(-1):

$P\left(-1\right)={\left(-1\right)}^{51}+51$

Since (-1) raised to any odd power is -1, we get:

$-1+51=50$

Thus, the remainder is 50.

Final answer: 50