If $(1 + 2!)$ is a root of the equation ${x^2} + bx + c = 0,$ where b and c are real then (b, c) is given by

Evaluating the expression 1 + 2! gives 1 + 2 = 3, so x = 3 is a root of the given equation x^2 + bx + c = 0. Substituting x = 3 into the equation yields 9 + 3b + c = 0. Checking the provided pair (2, -15) in this equation gives 9 + 3(2) + (-15) = 9 + 6 - 15 = 0. Therefore, the required values for (b, c) are (2, -15).

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