To solve the inequality $(x - 5)(x - 6) \le 56$: Step 1: Expand the left side of the inequality. $$ (x - 5)(x - 6) = x^2 - 6x - 5x + 30 = x^2 - 11x + 30 $$ Step 2: Rewrite the inequality with all terms on one side. $$ x^2 - 11x + 30 \le 56 $$ Subtract 56 from both sides: $$ x^2 - 11x + 30 - 56 \le 0 $$ $$ x^2 - 11x - 26 \le 0 $$ Step 3: Find the roots of the corresponding quadratic equation $x^2 - 11x - 26 = 0$. We can use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Here, $a=1$, $b=-11$, $c=-26$. The discriminant is $\Delta = b^2 - 4ac = (-11)^2 - 4(1)(-26) = 121 + 104 = 225$. The square root of the discriminant is $\sqrt{225} = 15$. The roots are: $$ x_1 = \frac{-(-11) - 15}{2(1)} = \frac{11 - 15}{2} = \frac{-4}{2} = -2 $$ $$ x_2 = \frac{-(-11) + 15}{2(1)} = \frac{11 + 15}{2} = \frac{26}{2} = 13 $$ So, the roots are $x = -2$ and $x = 13$. Step 4: Determine the intervals where the inequality $x^2 - 11x - 26 \le 0$ holds true. Since the coefficient of $x^2$ is positive ($a=1$), the parabola opens upwards. This means the quadratic expression is less than or equal to zero between its roots, inclusive. Therefore, the solution to the inequality is: $$ -2 \le x \le 13 $$ The final answer is $\boxed{-2 \le x \le 13}$.