Let $P = a^{2} - b^{2} - 2 a b, Q = a^{2} + 4 b^{2} - 6 a b, R = b^{2} + 6, S = a^{2} - 4 a b$ and $T = - 2 a^{2} + b^{2} - a b + a$ .Find the value of $P + Q + R + S + T$ ?

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Solution

$P + Q = a^{2} - b^{2} - 2 a b + a^{2} + 4 b^{2} - 6 a b = 2 a^{2} - 8 a b + 3 b^{2}$

$P + Q + R = 2 a^{2} - 8 a b + 3 b^{2} + b^{2} + 6 = 2 a^{2} - 8 a b + 4 b^{2} + 6$

$P + Q + R + S = 2 a^{2} - 8 a b + 4 b^{2} + 6 + a^{2} - 4 a b = 3 a^{2} - 12 a b + 4 b^{2} + 6$

$P + Q + R + S + T = 3 a^{2} - 12 a b + 4 b^{2} + 6 - 2 a^{2} + b^{2} - a b + a = a^{2} - 12 a b + 5 b^{2} + 6 - a b + a$

$∴ P + Q + R + S + T = a^{2} + 5 b^{2} - 12 a b - a b + a + 6$

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