To solve the quadratic equation \( x^2 + 10x + 25 = 0 \) using the perfect square trinomial pattern, we first recognize that the left-hand side can be expressed as a square of a binomial.
The expression \( x^2 + 10x + 25 \) is a perfect square trinomial because it can be rewritten as:
\[ (x + 5)^2 = 0 \]
Now, we set this equation to zero:
\[ (x + 5)^2 = 0 \]
To solve for \( x \), we take the square root of both sides:
\[ x + 5 = 0 \]
Subtracting 5 from both sides, we find:
\[ x = -5 \]
Thus, the solution to the equation \( x^2 + 10x + 25 = 0 \) is
\[ \boxed{-5} \]
Since this is a perfect square, it has only one solution (a double root).