Answer
(5z+1)(z+3)
Work Step by Step
In order to do factoring when the first coefficient is not one, we must use some trial and error. After all, when considering the factor $(ax +b)(cx +d)$, a and c are no longer simply one. Thus, in addition to considering all of the factors of the constant term (the term that is not multiplied by a variable), we must consider the factors of the coefficient for the squared term. Using all of these factors, it is then just a matter of doing trial and error until one of the combinations of factors multiplies to get the correct answer, which must multiply out to get the original expression.
We know that the factors of $5$ are: $5, 1$.
Using this information as well as guess and check, it follows that the factored form of the given expression is:
$$(5z+1)(z+3)$$
