Answer
$(a-6b)(a+9b)$
Work Step by Step
To factor a trinomial in the form $x^2+bx+c$, we must find two numbers whose product is $c$ and whose sum is $b$. We then insert these two numbers into the blanks of the factors $(x+\_)(x+\_)$. [In the case of this problem, the main variable is $a$, not $x$]
In the case of $a^2+3ab-54b^2$, we are looking for two numbers whose product is $-54b^2$ and whose sum is $3b$. The numbers $9b$ and $-6b$ meet these criteria, because $$9b\times-6b=-54b^2\;\text{and}\;9b+(-6b)=3b$$When we insert these numbers into the blanks, we arrive at the factors $(a-6b)(a+9b)$.
