1. **State the problem:** Write the quadratic expression $3x^2 - 24x + 54$ in the form $r(x + p)^2 + q$ where $r$, $p$, and $q$ are integers. 2. **Recall the formula:** The vertex form of a quadratic is $$r(x + p)^2 + q$$ where $r$ is the coefficient of $x^2$, and $p$, $q$ are constants to be found by completing the square. 3. **Start with the given expression:** $$3x^2 - 24x + 54$$ 4. **Factor out the coefficient of $x^2$ from the first two terms:** $$3(x^2 - 8x) + 54$$ 5. **Complete the square inside the parentheses:** Take half of the coefficient of $x$, which is $-8$, half is $-4$, then square it: $$(-4)^2 = 16$$ 6. **Add and subtract 16 inside the parentheses to complete the square:** $$3(x^2 - 8x + 16 - 16) + 54$$ 7. **Rewrite as:** $$3((x - 4)^2 - 16) + 54$$ 8. **Distribute the 3:** $$3(x - 4)^2 - 48 + 54$$ 9. **Simplify the constants:** $$3(x - 4)^2 + 6$$ 10. **Rewrite in the form $r(x + p)^2 + q$:** Note that $(x - 4)^2 = (x + (-4))^2$, so $p = -4$. **Final answer:** $$3(x + (-4))^2 + 6$$ or simply $$3(x - 4)^2 + 6$$