$$(2x+1)^3 = 0$$
Answer
Explanation
$ \color{blue}{ 8x^{3}+12x^{2}+6x+1 } $ is a polynomial of degree 3. To find zeros for polynomials of degree 3 or higher we use Rational Root Test.
The Rational Root Theorem tells you that if the polynomial has a rational zero then it must be a fraction $ \dfrac{p}{q} $, where p is a factor of the trailing constant and q is a factor of the leading coefficient.
The factors of the leading coefficient ( 8 ) are 1 2 4 8 .The factor of the constant term ( 1 ) is 1 . Then the Rational Roots Tests yields the following possible solutions:
$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 1 }{ 2 } , ~ \pm \frac{ 1 }{ 4 } , ~ \pm \frac{ 1 }{ 8 } ~ $$Substitute the POSSIBLE roots one by one into the polynomial to find the actual roots. Start first with the whole numbers.
If we plug these values into the polynomial $ P(x) $, we obtain $ P(-\frac{ 1 }{ 2 }) = 0 $.
To find remaining zeros we use Factor Theorem. This theorem states that if $\frac{p}{q}$ is root of the polynomial then this polynomial can be divided with $ \color{blue}{q x - p} $. In this example:
Divide $ P(x) $ with $ \color{blue}{ 2 x + 1 } $
$$ \frac{ 8x^{3}+12x^{2}+6x+1 }{ \color{blue}{ 2x + 1 } } = 4x^{2}+4x+1 $$Polynomial $ 4x^{2}+4x+1 $ can be used to find the remaining roots.
$ \color{blue}{ 4x^{2}+4x+1 } $ is a second degree polynomial. For a detailed answer how to find its roots you can use step-by-step quadratic equation solver.