Multiply each term of $ \left( \color{blue}{z+3}\right) $ by each term in $ \left( z+4\right) $.

$$ \left( \color{blue}{z+3}\right) \cdot \left( z+4\right) = z^2+4z+3z+12 $$

Combine like terms:

$$ z^2+ \color{blue}{4z} + \color{blue}{3z} +12 = z^2+ \color{blue}{7z} +12 $$

Subtract $ \dfrac{3}{z+3} $ from $ \dfrac{z+6}{z^2+7z+12} $ to get $ \dfrac{ \color{purple}{ -2z-6 } }{ z^2+7z+12 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{z+4}$.

$$ \begin{aligned} \frac{z+6}{z^2+7z+12} - \frac{3}{z+3} & = \frac{ z+6 }{ z^2+7z+12 } - \frac{ 3 \cdot \color{blue}{ \left( z+4 \right) }}{ \left( z+3 \right) \cdot \color{blue}{ \left( z+4 \right) }} = \\[1ex] &=\frac{ \color{purple}{ z+6 } }{ z^2+7z+12 } - \frac{ \color{purple}{ 3z+12 } }{ z^2+7z+12 }=\frac{ \color{purple}{ z+6 - \left( 3z+12 \right) } }{ z^2+7z+12 } = \\[1ex] &=\frac{ \color{purple}{ -2z-6 } }{ z^2+7z+12 } \end{aligned} $$

Simplify $ \dfrac{-2z-6}{z^2+7z+12} $ to $ \dfrac{-2}{z+4} $.

Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{z+3}$.

$$ \begin{aligned} \frac{-2z-6}{z^2+7z+12} & =\frac{ \left( -2 \right) \cdot \color{blue}{ \left( z+3 \right) }}{ \left( z+4 \right) \cdot \color{blue}{ \left( z+3 \right) }} = \\[1ex] &= \frac{-2}{z+4} \end{aligned} $$