1. **State the problem:** Factorize the expression $x^2 + 25y^2$. 2. **Recall the formula:** The sum of squares $a^2 + b^2$ generally cannot be factorized over the real numbers into linear factors. It can be factorized over complex numbers as $a^2 + b^2 = (a + bi)(a - bi)$. 3. **Apply to the problem:** Here, $a = x$ and $b = 5y$, so: $$x^2 + 25y^2 = x^2 + (5y)^2$$ 4. **Factorization over complex numbers:** $$x^2 + 25y^2 = (x + 5iy)(x - 5iy)$$ 5. **Explanation:** Since this is a sum of squares, it cannot be factorized into real linear factors. The factorization involves imaginary unit $i = \sqrt{-1}$. **Final answer:** $$x^2 + 25y^2 = (x + 5iy)(x - 5iy)$$