To solve the quadratic equation \(x^2 + 10x + 25 = 0\) using the perfect square trinomial pattern, we first recognize that this expression can be factored.
The perfect square trinomial can be expressed in the form \((x + a)^2\), where \(a\) is a number.
In this case, we notice that:
\[ x^2 + 10x + 25 = (x + 5)^2 \]
This is because \(5^2 = 25\) and \(2 \cdot 5 = 10\).
Now our equation becomes:
\[ (x + 5)^2 = 0 \]
Next, we can take the square root of both sides:
\[ x + 5 = 0 \]
Subtracting 5 from both sides gives us:
\[ x = -5 \]
Thus, the solution to the equation \(x^2 + 10x + 25 = 0\) is:
\[ \boxed{-5} \]