For how many integer values of m does the equation $x^{2} - 10 x + (m + 3) (m - 4) = 0$ have roots of opposite signs?

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Solution

If the roots are of opposite signs, product of roots must be negative.
i.e. $(m + 3) (m - 4) < 0 ⟹ - 3 < m < 4$
Number of integer values satisfying the above inequality is 6 (-2, -1, 0, 1, 2, 3).

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