1. **State the problem:** Solve the expression $$xy(k^2+1) + k(x^2 + y^2)$$ for simplification or factoring. 2. **Identify the formula and rules:** We want to simplify or factor the expression. Notice that it involves terms with $xy$, $x^2$, and $y^2$. 3. **Rewrite the expression:** $$xy(k^2+1) + k(x^2 + y^2) = xyk^2 + xy + kx^2 + ky^2$$ 4. **Group terms:** Group terms to see if factoring is possible: $$kx^2 + ky^2 + xyk^2 + xy$$ 5. **Factor by grouping:** Group as: $$k(x^2 + y^2) + xy(k^2 + 1)$$ This is the original expression, so no further factoring by grouping. 6. **Check for special factorizations:** Recall that $x^2 + y^2$ and $xy$ do not factor nicely over the reals unless combined as $(x+y)^2 = x^2 + 2xy + y^2$ or $(x-y)^2 = x^2 - 2xy + y^2$. 7. **Try expressing in terms of $(x+y)^2$ and $xy$:** Note that: $$x^2 + y^2 = (x+y)^2 - 2xy$$ Substitute: $$k(x^2 + y^2) + xy(k^2 + 1) = k((x+y)^2 - 2xy) + xy(k^2 + 1) = k(x+y)^2 - 2kxy + k^2xy + xy$$ Simplify the $xy$ terms: $$k(x+y)^2 + xy(k^2 + 1 - 2k)$$ Note that: $$k^2 + 1 - 2k = (k-1)^2$$ So the expression becomes: $$k(x+y)^2 + xy(k-1)^2$$ 8. **Final simplified form:** $$\boxed{k(x+y)^2 + xy(k-1)^2}$$ This is a neat factorization showing the expression in terms of squares. **Answer:** $$k(x+y)^2 + xy(k-1)^2$$