Solve the quadratic equation : $(m^{2} + m) (m^{2} + m - 3) = 28$

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Solution

Let $m^{2} + m = k$
$k (k - 3) = 28$
$⟹ k^{2} - 3 k - 28 = 0$
$⟹ (k - 7) (k + 4) = 0$
If $k - 7 = 0$ :
$m^{2} + m - 7 = 0$
$⟹ m = \frac{- 1 \pm \sqrt{29}}{2}$
If $k + 4 = 0$ :
$m^{2} + m + 4 = 0$
The above has no real solutions as discriminant is negative.
Therefore, final solution:
$m = \frac{- 1 \pm \sqrt{29}}{2}$

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