### Instruction: If
\[x^5 + 5x^4 + 10x^3 + 10x^2 - 5x + 1 = 10,\]and $x \neq -1,$ compute the numerical value of $(x + 1)^4.$.### Response: I notice that the left-hand side of the equation has a familiar pattern: it is the expansion of $(x + 1)^5$ by the binomial theorem.
So I can rewrite the equation as $(x + 1)^5 = 10.$
Taking the fifth root of both sides, I get $x + 1 = \sqrt[5]{10}.$
Now I just need to raise both sides to the fourth power to get the desired expression.
So $(x + 1)^4 = (\sqrt[5]{10})^4 = 10^{4/5}.$