Do not use a calculator in answering this question.

1. Identify the complex conjugate root.

Since the coefficients of the polynomial $x^3 + ax^2 + 18x + b = 0$ are real, the complex roots must occur in conjugate pairs.
Therefore, another root is $x = 1 - 3\mathrm{i}$.

2. Form the quadratic factor corresponding to the complex roots.

The quadratic factor formed by the conjugate pair is:
$(x - (1 + 3\mathrm{i}))(x - (1 - 3\mathrm{i}))$
$= ((x - 1) - 3\mathrm{i})((x - 1) + 3\mathrm{i})$
$= (x - 1)^2 - (3\mathrm{i})^2$
$= (x^2 - 2x + 1) - 9(-1)$
$= x^2 - 2x + 10$

3. Express the cubic polynomial in factored form and compare coefficients to find the third root.

Let the third root be the real number $\alpha$. We can express the cubic polynomial as the product of the quadratic factor and $(x - \alpha)$:
$x^3 + ax^2 + 18x + b = (x^2 - 2x + 10)(x - \alpha)$
$= x^3 - \alpha x^2 - 2x^2 + 2\alpha x + 10x - 10\alpha$
$= x^3 - (\alpha + 2)x^2 + (2\alpha + 10)x - 10\alpha$
Comparing the coefficients of $x$:
$18 = 2\alpha + 10$
$2\alpha = 8 \implies \alpha = 4$
The third root is $4$.

4. Compare the remaining coefficients to find $a$ and $b$.

Comparing the coefficients of $x^2$:
$a = -(\alpha + 2)$
$a = -(4 + 2) = -6$

Comparing the constant terms:
$b = -10\alpha$
$b = -10(4) = -40$