Find the product.(i) (a + b) (c − d) + (a − b) (c + d) + 2 (ac + b

1.	Find the product.(i) (a + b) (c − d) + (a − b) (c + d) + 2 (ac + bd)(ii) (x + y) (2x + y) + (x + 2y) (x − y)
Answer» (i) (a + b) (c − d) + (a − b) (c + d) + 2 (ac + bd) = a (c − d) + b (c − d) + a (c + d) − b (c + d) + 2 (ac + bd) = ac − ad + bc − bd + ac + ad − bc − bd + 2ac + 2bd = (ac + ac + 2ac) + (ad − ad) + (bc − bc) + (2bd − bd − bd) = 4ac (ii) (x + y) (2x + y) + (x + 2y) (x − y) = x (2x + y) + y (2x + y) + x (x − y) + 2y (x − y) = 2x² + xy + 2xy + y² + x² − xy + 2xy − 2y² = (2x² + x²) + (y² − 2y²) + (xy + 2xy − xy + 2xy) = 3x² − y² + 4xy

Find the product.(i) (a + b) (c − d) + (a − b) (c + d) + 2 (ac + bd)(ii) (x + y) (2x + y) + (x + 2y) (x − y)

Answer»

(i) (a + b) (c − d) + (a − b) (c + d) + 2 (ac + bd)

= a (c − d) + b (c − d) + a (c + d) − b (c + d) + 2 (ac + bd)

= ac − ad + bc − bd + ac + ad − bc − bd + 2ac + 2bd

= (ac + ac + 2ac) + (ad − ad) + (bc − bc) + (2bd − bd − bd)

= 4ac

(ii) (x + y) (2x + y) + (x + 2y) (x − y)

= x (2x + y) + y (2x + y) + x (x − y) + 2y (x − y)

= 2x² + xy + 2xy + y² + x² − xy + 2xy − 2y²

= (2x² + x²) + (y² − 2y²) + (xy + 2xy − xy + 2xy)

= 3x² − y² + 4xy