Factor Polynomial Ca4132

1. **State the problem:** Factor the polynomial $$x^4 + 4x^2 + 3$$ completely over the integers. 2. **Recognize the form:** Notice that the polynomial is quadratic in terms of $x^2$. Let $y = x^2$, then the polynomial becomes $$y^2 + 4y + 3$$. 3. **Factor the quadratic in $y$:** We look for two numbers that multiply to $3$ and add to $4$. These numbers are $1$ and $3$. 4. **Write the factorization:** $$y^2 + 4y + 3 = (y + 1)(y + 3)$$. 5. **Substitute back $y = x^2$:** $$ (x^2 + 1)(x^2 + 3) $$. 6. **Check for further factorization:** Both $x^2 + 1$ and $x^2 + 3$ have no real integer factors since their discriminants are negative. 7. **Final answer:** The complete factorization over the integers is $$\boxed{(x^2 + 1)(x^2 + 3)}$$.