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.
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.