Final answer:
The equation (x + 2)^4 = x^4 + 16 is not true for all real numbers x because when expanded, (x + 2)^4 includes additional terms that do not equal zero. The correct answer is option (C) some, since there may be specific cases where the equation is true.
Step-by-step explanation:
To solve the given question, we need to determine if the equation (x + 2)^4 = x^4 + 16 holds true for all, no, some, or none of the above real numbers x. To do this, we should try to simplify or expand both sides to see if they are equivalent. On the left-hand side, use the binomial theorem or distribution to expand (x + 2)^4, which results in the expression x^4 + 8x^3 + 24x^2 + 32x + 16. Comparing it to the right-hand side of the equation, which is x^4 + 16, indicates that the equation will generally not hold for all values of x because the expanded form includes additional terms like 8x^3, 24x^2, and 32x that are not canceled out or equal to zero for all real numbers. Thus, the correct answer for which real numbers the equation holds true is option (C) some, as there may be specific values of x for which the equation could be true (specifically when x=0).