The expression $x^2 + 15x + 54$ can be written as  $(x + a)(x + b),$ and the expression $x^2 - 17x + 72$ written as $(x - b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$?
To write a quadratic expression as a product of two linear factors, I need to find two numbers that add up to the coefficient of $x$ and multiply to the constant term. For $x^2 + 15x + 54$, I see that $6$ and $9$ are such numbers, since $6 + 9 = 15$ and $6 \cdot 9 = 54$. So I can write $x^2 + 15x + 54$ as $(x + 6)(x + 9)$. That means $a = 6$ and $b = 9$. For $x^2 - 17x + 72$, I see that $8$ and $9$ are such numbers, since $8 + 9 = 17$ and $8 \cdot 9 = 72$. So I can write $x^2 - 17x + 72$ as $(x - 9)(x - 8)$. That means $b = 9$ and $c = 8$. To find $a + b + c$, I just add up these values: $6 + 9 + 8 = 23$. 