1.Multiply the binomials:
(i) \((2x + 5)\)and \((4x – 3)\)

(ii) \((y – 8)\) and \((3y – 4)\)

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

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

(v) \((2pq + 3q^2)\) and \((3pq – 2q^2)\)

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

(i)Multiplying the given expressions as:\( (2x + 5) \times (4x – 3)\)

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

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

\(= 8x^2 – 6x + 20x – 15\)

\(= 8x^2 + 14x – 15\)

(ii)Multiplying the given expressions as: \((y – 8) \times (3y – 4)\)

\(= y \times (3y – 4) – 8 \times (3y – 4)\)

\(= (y \times 3y) – (y \times 4) – (8 \times 3y) + (-8 \times -4)\)

\(= 3y^2 – 4y – 24y + 32\)

\(= 3y^2 – 28y + 32\)

(iii)Multiplying the given expressions as: \((2.5l – 0.5m) × (2.5l + 0.5m)\)

\(= (2.5l \times 2.5l) + (2.5l \times 0.5m) – (0.5m \times 2.5l) – (0.5m \times 0.5m)\)

\(= 6.25l^2 + 1.25ml – 1.25ml – 0.25m^2\)

\(= 6.25l^2 + 0 – 0.25m^2\)

\(= 6.25l^2 – 0.25m^2\)

(iv)Multiplying the given expressions as: \((a + 3b) \times (x + 5)\)

\(= a \times (x + 5) + 36 \times (x + 5)\)

\(= (a \times x) + (a \times 5) + (36 \times x) + (36 \times 5)\)

\(= ax + 5a + 3bx + 15b\)

(v) Multiplying the given expressions as:\((2pq + 3q^2) \times (3pq – 2q^2)\)

\(= 2pq \times (3pq – 2q^2) + 3q^2 (3pq – 2q^2)\)

\(= (2pq \times 3pq) – (2pq \times 2q^2) + (3q^2 × 3pq) – (3q^2 \times 2q^2)\)

\(= 6p^2q^2 – 4pq^3 + 9pq^3 – 6q^4\)

\(= 6p^2q^2 + 5pq^3 – 6q^4\)

(vi)Multiplying the given expressions as:\((\frac { 3 }{ 4 }a^2 + 3b^2) \times 4(a^2 – \frac { 2 }{ 3 } b^2)\)

\(=(\frac{3}4a^2+3b^2)\times(4a^2-\frac{8}3b^2)\)

\(=\frac{3}4a^2\times (4a^2-\frac{8}3b^2)+3b^2\times(4a^2-\frac{8}3b^2)\)

\(=(\frac{3}4a^2\times 4b^2) - (\frac{3}4a^2\times \frac{8}3b^2) + (3b^2\times 4a^2)-(3b^2\times\frac{8}3b^2)\)

\(=3a^4-2a^2b^2+12a^2b^2-8b^4\)

\(=3a^4+10a^2b^2-8b^4\)