Move all terms to the left-hand side to factor: $18(w - a)(w + 5) - 6(w + 5) = 0$ $(w + 5)[18(w - a) - 6] = 0$

Setting each factor to zero gives the two solutions:

1. $w + 5 = 0 \implies w_1 = -5$
2. $18(w - a) - 6 = 0 \implies 18(w - a) = 6 \implies w - a = \dfrac{1}{3} \implies w_2 = a + \dfrac{1}{3}$

Step 2: Use the given sum of the solutions

We are given that the sum of the solutions is $\dfrac{37}{3}$: $w_1 + w_2 = \dfrac{37}{3}$ $-5 + \left(a + \dfrac{1}{3}\right) = \dfrac{37}{3}$ $a - \dfrac{14}{3} = \dfrac{37}{3}$

Step 3: Solve for $a$

Add $\dfrac{14}{3}$ to both sides: $a = \dfrac{37}{3} + \dfrac{14}{3} = \dfrac{51}{3} = 17$

Thus, the value of $a$ is 17.