Multiply the inequality by -1 to make the coefficient of the highest power in $a-3a^{2}-3$ positive. Since $-1$ is negative, the inequality direction is changed.
$$-a+3a^{2}+3\geq 0$$
To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$, where $x_{1}$ and $x_{2}$ are the solutions of the quadratic equation $ax^{2}+bx+c=0$.
$$-a+3a^{2}+3=0$$
All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. Substitute $3$ for $a$, $-1$ for $b$, and $3$ for $c$ in the quadratic formula.
Since the square root of a negative number is not defined in the real field, there are no solutions. Expression $-a+3a^{2}+3$ has the same sign for any $a$. To determine the sign, calculate the value of the expression for $a=0$.
$$-0+3\times 0^{2}+3=3$$
The value of the expression $-a+3a^{2}+3$ is always positive. Inequality holds for $a\in \mathrm{R}$.
