Let's first expand the terms on both sides of the inequality:

(x-3)² = x² - 6x + 9
(5+x)² = x² + 10x + 25

Now, our inequality becomes:

x² - 6x + 9 ≤ x² + 10x + 25

Subtract x² from both sides to simplify:

-6x + 9 ≤ 10x + 25

Subtract 10x from both sides:

-16x + 9 ≤ 25

Subtract 9 from both sides:

-16x ≤ 16

Divide by -16 (Note: when dividing by a negative number, the direction of the inequality changes):

x ≥ -1

Therefore, the solution to the inequality (x-3)² ≤ (5+x)² is x ≥ -1.