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\frac{x^{4}}{4}+\frac{4\left(x^{2}+1\right)}{4}
To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2}+1 times \frac{4}{4}.
\frac{x^{4}+4\left(x^{2}+1\right)}{4}
Since \frac{x^{4}}{4} and \frac{4\left(x^{2}+1\right)}{4} have the same denominator, add them by adding their numerators.
\frac{x^{4}+4x^{2}+4}{4}
Do the multiplications in x^{4}+4\left(x^{2}+1\right).
\frac{x^{4}+4x^{2}+4}{4}
Factor out \frac{1}{4}.
\left(x^{2}+2\right)^{2}
Consider x^{4}+4x^{2}+4. Use the perfect square formula, a^{2}+2ab+b^{2}=\left(a+b\right)^{2}, where a=x^{2} and b=2.
\frac{\left(x^{2}+2\right)^{2}}{4}
Rewrite the complete factored expression. Polynomial x^{2}+2 is not factored since it does not have any rational roots.