What must be added to $4 x^{2} + 12 x + 5$ to make it a perfect square?

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Solution

$4 x^{2} + 12 x + 5 = 4 [x^{2} + 3 x + (\frac{5}{4})] = 4 [(x + (\frac{3}{2})^{2} - (\frac{3}{2})^{2} + (\frac{5}{4}))] = 4 [(x + (\frac{3}{2})^{2} - (\frac{4}{4}))] = 4 (x + (\frac{3}{2})^{2} - 4) h e n c e 4 m u s t b e a d d e d t o m a k e i t p e r f e c t s q u a r e .$

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