Simplify the following:

$(i) (1.5 x - 4 y) (1.5 x + 4 y + 3)$

$(i i) (x + y) (2 x + y) + (x + 2 y) (x - y)$

$(i i i) (a^{2} + 5) (b^{3} + 3) + 5$

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Solution

$(i) (1.5 x - 4 y) (1.5 x + 4 y + 3)$

$= (1.5 x \times 1.5 x) + (1.5 x \times 4 y) + (1.5 x \times 3) - (4 y \times 1.5 x) - (4 y \times 4 y) - (4 y \times 3)$

$= 2.25 x^{2} + 6 x y + 4.5 x - 6 x y - 16 y^{2} - 12 y$

$= 2.25 x^{2} - 16 y^{2} + 4.5 x - 12 y$

$(i i) (x + y) (2 x + y) + (x + 2 y) (x - y)$

$= (x \times 2 x) + (x \times y) + (y \times 2 x) + (y \times y) + (x \times x) - (x \times y) + (2 y \times x) - (2 y \times y)$

$= 2 x^{2} + x y + 2 x y + y^{2} + x^{2} + x y - 2 y^{2}$

$= 3 x^{2} + 4 x y - y^{2}$

$(i i i) (a^{2} + 5) (b^{3} + 3) + 5$

$= (a^{2} \times b^{3} + a^{2} \times 3 + 5 \times b^{3} + 5 \times 3) + 5$

$= a^{2} b^{3} + 3 a^{2} + 5 b^{3} + 15 + 5$

$= a^{2} b^{3} + 3 a^{2} + 5 b^{3} + 20$

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