$$(x+3)(x+5)(x+7)(x+9)-9 = 0$$
$$ \begin{matrix}x_1 = -6 & x_2 = -6-\sqrt{ 10 } & x_3 = -6+\sqrt{ 10 } \end{matrix} $$
$ \color{blue}{ x^{4}+24x^{3}+206x^{2}+744x+936 } $ is a polynomial of degree 4. 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 factor of the leading coefficient ( 1 ) is 1 .The factors of the constant term (936) are 1 2 3 4 6 8 9 12 13 18 24 26 36 39 52 72 78 104 117 156 234 312 468 936 . Then the Rational Roots Tests yields the following possible solutions:
$$ \pm \frac{ 1 }{ 1 } , ~ \pm \frac{ 2 }{ 1 } , ~ \pm \frac{ 3 }{ 1 } , ~ \pm \frac{ 4 }{ 1 } , ~ \pm \frac{ 6 }{ 1 } , ~ \pm \frac{ 8 }{ 1 } , ~ \pm \frac{ 9 }{ 1 } , ~ \pm \frac{ 12 }{ 1 } , ~ \pm \frac{ 13 }{ 1 } , ~ \pm \frac{ 18 }{ 1 } , ~ \pm \frac{ 24 }{ 1 } , ~ \pm \frac{ 26 }{ 1 } , ~ \pm \frac{ 36 }{ 1 } , ~ \pm \frac{ 39 }{ 1 } , ~ \pm \frac{ 52 }{ 1 } , ~ \pm \frac{ 72 }{ 1 } , ~ \pm \frac{ 78 }{ 1 } , ~ \pm \frac{ 104 }{ 1 } , ~ \pm \frac{ 117 }{ 1 } , ~ \pm \frac{ 156 }{ 1 } , ~ \pm \frac{ 234 }{ 1 } , ~ \pm \frac{ 312 }{ 1 } , ~ \pm \frac{ 468 }{ 1 } , ~ \pm \frac{ 936 }{ 1 } ~ $$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(-6) = 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}{x + 6} $
$$ \frac{ x^{4}+24x^{3}+206x^{2}+744x+936 }{ \color{blue}{ x + 6 } } = x^{3}+18x^{2}+98x+156 $$Polynomial $ x^{3}+18x^{2}+98x+156 $ can be used to find the remaining roots.
Use the same procedure to find roots of $ x^{3}+18x^{2}+98x+156 $
When you get second degree polynomial use step-by-step quadratic equation solver to find two remaining roots.