(a^2+cb^2)(d^2+ce^2)=(ad+cbe)^2+c(ae-bd)^2

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вопрос

This is a well-known identity in algebra. To prove it, we expand the left-hand side of the equation:

(a^2 + cb^2)(d^2 + ce^2)
= a^2d^2 + a^2ce^2 + cb^2d^2 + cb^2ce^2
= a^2d^2 + a^2ce^2 + b^2cde + b^2ce^2
= a^2d^2 + a^2ce^2 + b^2cde + b^2ce^2

Now, we can group the terms to simplify further:

= a^2d^2 + b^2cde + a^2ce^2 + b^2ce^2
= d^2(a^2 + b^2c) + ce^2(a^2 + b^2c)
= (d^2 + ce^2)(a^2 + b^2c)

Therefore, (a^2+cb^2)(d^2+ce^2) = (ad+cbe)^2 + c(ae-bd)^2, as required.