$4 a^{2} + b^{2} + 4 a b + 8 a + 4 b + 4$ =?

(a) $(2 a + b + 2)^{2}$

(b) $(2 a - b + 2)^{2}$

(c) $(a + 2 b + 2)^{2}$

(d) None of these

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Solution

(a) $(2 a + b + 2)^{2}$

$4 a^{2} + b^{2} + 4 a b + 8 a + 4 b + 4 = 4 a^{2} + b^{2} + 4 + 4 a b + 4 b + 8 a = 2 a^{2} + b^{2} + 2^{2} + 2 \times 2 a \times b + 2 \times b \times 2 + 2 \times 2 a \times 2 = (2 a + b + 2)^{2}$

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