Expand the product ${(x+5)(x+7)}$.
When using the distributive property for the first time, we add the product of $x+5$ and $x$ to the product of $x+5$ and 7: \begin{align*}
(x+5)(x+7) &= (x+5) \cdot x + (x+5) \cdot 7\\
&= x(x+5) + 7(x+5).
\end{align*}We use the distributive property again and combine like terms: \begin{align*}
x(x+5) + 7(x+5) &= x^2 + 5x + 7x+ 35\\
&= \boxed{x^2 + 12x + 35}.
\end{align*}