Multiply the binomials.

(i) $(2x + 5) \text{and} (4x – 3)$

(ii) $(y – 8) \text{and} (3y – 4)$

(iii) $(2.5l – 0.5m) \text{and} (2.5l + 0.5m)$

(iv) $(a + 3b) \text{and} (x + 5)$

(v) $(2pq + 3q^2)\text{ and} (3pq – 2q^2)$

(vi) $\left(\frac{3}{4}{a}^2 + 3b^2\right) \text{and} 4\left(a^2 – \frac{2}{3} b^2\right)$

(i) Consider the given binomial $(2x + 5) \text{and} (4x – 3)$.

Perform multiplication to get the following,

$(2x + 5)(4x – 3) =2x\times 4x–2x\times 3+5\times 4x–5\times 3$

$=8x²–6x+20x-15$

$=8x²+14x-15$

Hence, the simplification of the given binomial is $8x²+14x-15$.

(ii) Consider the given binomial $(y – 8) \text{and} (3y – 4)$.

Perform multiplication to get the following,

$(y-8)(3y-4) =y \times 3y-4y-8 \times 3y+32$

$=3y^2-4y-24y+32$

$=3y^2-28y+32$

Hence, the simplification of the given binomial is $3y^2-28y+32$.

(iii) Consider the given binomial $(2.5l – 0.5m) \text{and} (2.5l + 0.5m)$.

Perform multiplication to get the following,

$(2.5l-0.5m)(2.5l+0.5m)=2.5l\times 2.5 l+2.5l\times 0.5m-0.5m\times 2.5l-0.5m\times 0.5m$

$= 6.25l^2+1.25 lm-1.25 lm-0.25 m^2$

$=6.25l^2-0.25 m^2$

Hence, the simplification of the given binomial is $6.25l^2-0.25 m^2$.

(iv) Consider the given binomial $(a + 3b) \text{and} (x + 5)$.

Perform multiplication to get the following,

$(a+3b) (x+5)=ax+5a+3bx+15b$

Hence, the simplification of the given binomial is $ax+5a+3bx+15b$.

(v) Consider the given binomial $(2pq + 3q^2)\text{ and} (3pq – 2q^2)$.

Perform multiplication to get the following,

$(2pq+3q^2) (3pq-2q^2)=2pq\times 3pq-2pq\times 2q^2+3q^2\times 3pq-3q^2\times 2q^2$

$=6p^2{q}^2- 4p{q}^3+ 9p{q}^3-6q^4$

$=6p^2{q}^2+5p{q}^3-6q^4$

Hence, the simplification of the given binomial is $6p^2{q}^2+5p{q}^3-6q^4$

(vi) Consider the given binomial $\left(\frac{3}{4}{a}^2 + 3b^2\right) \text{and} 4\left(a^2 – \frac{2}{3} b^2\right)$.

Perform multiplication to get the following,

$\left(\frac{3}{4}{a}^2 + 3b^2\right)\times \left(4a^2 – 4\times \frac{2}{3} b^2\right)=\frac{3}{4}{a}^2\times 4a^2-\frac{3}{4}{a}^2\times 4\times \frac{2}{3} b^2+3b^2\times 4a^2-3b^2\times 4\times \frac{2}{3} b^2$

$=3a^4-2a^2{b}^2+12a^2{b}^2-8b^4$

$=3a^4+10a^2{b}^2-8b^4$

Hence, the simplification of the given binomial is $3a^4+10a^2{b}^2-8b^4.$