2.Use the identity \((x + a)(x + b) = x^2 + (a + b)x + ab\) to find the following products.
(i) \((x + 3) (x + 7)\)

(ii) \((4x + 5)(4x + 1)\)

(iii)\( (4x – 5) (4x – 1)\)

(iv) \((4x + 5) (4x – 1)\)

(v) \((2x + 5y) (2x + 3y)\)

(vi) \((2a2 + 9) (2a2 + 5)\)

(vii) \((xyz – 4) (xyz – 2)\)

(i)We have: \((x + 3) (x + 7)\)

\(=x^2 + (3 + 7)x + 3\times 7\)

\(= x^2 +10x + 21\)

(ii)We have: \((4x + 5) (4x + 1)\)

\(=(4x)^2 + (5 + 1)(4x) + 5\times 1\)

\(= 16x^2 +24x + 5\)

(iii)We have: \((4x - 5) (4x - 1)\)

\(=(4x)^2 - (5 + 1)(4x) + (-5)\times(-1)\)

\(= 16x^2 -24x + 5\)

(iv)We have: \((4x + 5) (4x - 1)\)

\(=(4x)^2 + (5 - 1)(4x) + (5)\times(-1)\)

\(= 16x^2 +16x - 5\)

(v)We have: \((2x + 5y) (2x + 3y)\)

\(=(2x)^2 + (5y +3y)(2x) + (5y)\times (3y)\)

\(=(2x)^2 + (8y)(2x) + (5y)\times (3y)\)

\(= 4x^2 +16xy + 15y^2\)

(vi)We have: \((2a^2 + 9) (2a^2 + 5)\)

\(=(2a^2)^2 + (9+5)(2a^2) + (5)\times (9)\)

\(=(2a^2)^2 + (14)(2a^2) + 45\)

\(= 4a^4 +28a^2 + 45\)

(vii)We have: \((xyz - 4) (xyz - 2)\)

\(=(xyz)^2 - (4+2)(xyz) + (-4)\times (-2)\)

\(=(xyz)^2 - (6)(xyz) + 8\)

\(= x^2y^2z^2 -6xyz + 8\)