**1. Multiply the binomials.**

**(i) (2x + 5) and (4x – 3)**

**Solution:**

= (2x + 5) (4x – 3)

= 2x x 4x – 2x x 3 + 5 x 4x – 5 x 3

= 8x² – 6x + 20x -15

= 8x² + 14x -15

**(ii) (y – 8) and (3y – 4)**

**Solution:**

= (y – 8) (3y – 4)

= y x 3y – 4y – 8 x 3y + 32

= 3y^{2 }– 4y – 24y + 32

= 3y^{2} – 28y + 32

**(iii) (2.5l – 0.5m) and (2.5l + 0.5m)**

**Solution:**

= (2.5l – 0.5m) (2.5l + 0.5m)

= 2.5l x 2.5 l + 2.5l x 0.5m – 0.5m x 2.5l – 0.5m x 0.5m

= 6.25l^{2 }+ 1.25 lm – 1.25 lm – 0.25 m^{2}

= 6.25l^{2}– 0.25 m^{2}

**(iv) (a + 3b) and (x + 5)**

**Solution:**

= (a + 3b) (x + 5)

= ax + 5a + 3bx + 15b

**(v) (2pq + 3q ^{2}) and (3pq – 2q**

^{2})

**Solution:**

= (2pq + 3q^{2}) (3pq – 2q^{2})

= 2pq x 3pq – 2pq x 2q^{2} + 3q^{2} x 3pq – 3q^{2} x 2q^{2}

= 6p^{2}q^{2} – 4pq^{3} + 9pq^{3} – 6q^{4}

= 6p^{2}q^{2} + 5pq^{3} – 6q^{4}

**(vi) (3/4 a ^{2} + 3b^{2}) and 4(a^{2} – 2/3 b^{2})**

**Solution:**

= (3/4 a² + 3b²) and 4(a² – 2/3 b²)

= (3/4 a² + 3b²) x (4a² – 8/3 b²)

= 3/4 a² x (4a² – 8/3 b²) + 3b² x (4a² – 8/3 b²)

= 3/4 a² x 4a² -3/4 a² x 8/3 b² + 3b² x 4a² – 3b² x 8/3 b²

= 3a^{4} – 2a² b² + 12 a² b² – 8b^{4}

= 3a^{4} + 10a² b² – 8b^{4}

**2. Find the product.**

**(i) (5 – 2x) (3 + x)**

**Solution:**

= (5 – 2x) (3 + x)

= 5 (3 + x) – 2x (3 + x)

=15 + 5x – 6x – 2x^{2}

= 15 – x -2 x^{2}

**(ii) (x + 7y) (7x – y)**

**Solution:**

= (x + 7y) (7x – y)

= x(7x-y) + 7y (7x-y)

=7x^{2 }– xy + 49xy – 7y^{2}

= 7x^{2 }– 7y^{2} + 48xy

**(iii) (a ^{2}+ b) (a + b^{2})**

**Solution:**

= (a^{2}+ b) (a + b^{2})

= a^{2}(a + b^{2}) + b(a + b^{2})

= a^{3} + a^{2}b^{2} + ab + b^{3}

= a^{3} + b^{3} + a^{2}b^{2} + ab

**(iv) (p ^{2} – q^{2}) (2p + q)**

**Solution:**

= (p^{2}– q^{2}) (2p + q)

= p^{2} (2p + q) – q^{2} (2p + q)

=2p^{3} + p^{2}q – 2pq^{2} – q^{3}

= 2p^{3} – q^{3} + p^{2}q – 2pq^{2}

**3. Simplify.**

**(i) (x ^{2}– 5) (x + 5) + 25**

**Solution:**

= (x^{2}– 5) (x + 5) + 25

= x^{3} + 5x^{2 }– 5x – 25 + 25

= x^{3} + 5x^{2} – 5x

**(ii) (a ^{2}+ 5) (b^{3}+ 3) + 5**

**Solution:**

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

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

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

**(iii)(t + s ^{2})(t^{2} – s)**

**Solution:**

= (t + s^{2}) (t^{2} – s)

= t (t^{2} – s) + s^{2}(t^{2} – s)

= t^{3 }– st + s^{2}t^{2} – s^{3}

= t^{3} – s^{3} – st + s^{2}t^{2}

**(iv) (a + b) (c – d) + (a – b) (c + d) + 2 (ac + bd)**

**Solution:**

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

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

= ac – ad + bc – bd + ac + ad – bc – bd + 2ac + 2bd

= 4ac

**(v) (x + y)(2x + y) + (x + 2y)(x – y)**

**Solution:**

= (x + y)(2x + y) + (x + 2y)(x – y)

= 2x^{2} + xy + 2xy + y^{2} + x^{2} – xy + 2xy – 2y^{2}

= 3x^{2} + 4xy – y^{2}

**(vi) (x + y) (x ^{2}– xy + y^{2})**

**Solution:**

= (x + y) (x^{2}– xy + y^{2})

= x^{3} – x^{2}y + xy^{2} + x2y – xy^{2} + y^{3}

= x^{3} + y^{3}

**(vii) (1.5x – 4y) (1.5x + 4y + 3) – 4.5x + 12y**

**Solution:**

= (1.5x – 4y) (1.5x + 4y + 3) – 4.5x + 12y

= 2.25x^{2} + 6xy + 4.5x – 6xy – 16y^{2} – 12y – 4.5x + 12y

= 2.25x^{2} – 16y^{2}

**(viii) (a + b + c) (a + b – c)**

**Solution:**

= (a + b + c) (a + b – c)

= a^{2} + ab – ac + ab + b^{2 }– bc + ac + bc – c^{2}

= a^{2} + b^{2} – c^{2} + 2ab

**👍👍👍**