4.Simplify:
(i)\( (a^2 – b^2)^2\)

(ii) \((2x + 5)^2 – (2x – 5)^2\)

(iii) \((7m – 8n)^2 + (7m + 8n)^2\)

(iv) \((4m + 5n)^2 + (5m + 4n)^2\)

(v) \((2.5p – 1.5q)^2 – (1.5p – 2.5q)^2\)

(vi) \((ab + bc)^2 – 2ab^2c\)

(vii) \((m^2 – n^2m)^2 + 2m^3n^2\)

(i)Given:\((a^2-b^2)^2\)
\(=(a^2)^2-2a^2b^2+(b^2)^2\quad\)[use \((a-b)^2=a^2-2ab+b^2\)]

\(=a^4-2a^2b^2+b^4\)

(ii)Given:\((2x + 5)^2 – (2x – 5)^2\)

\(=[(2x)^2+2(2x)(5)+(5)^2]-[(2x)^2-2(2x)(5)+(5)^2]\)

\(=[4x^2+20x+25]-[4x^2-20x+25]\)

\(=4x^2+20x+25-4x^2+20x-25]\)

\(=(4x^2-4x^2)+(+20x+20x)+(+25-25)\quad \)[Placing same terms together]

\(=40x\)

(iii)Given:\((7m – 8n)^2 + (7m + 8n)^2\)

\(=[(7m)^2-2(7m)(8n)+(8n)^2] + [(7m)^2+2(7m)(8n)+(8n)^2]\)

\(=[49m^2-112mn+64n^2]+[49m^2+112mn+64n^2]\)

\(=(49m^2+49m^2)+(-112mn+112mn)+(64n^2+64n^2)\quad \)[Placing same terms together]

\(=98m^2+128n^2\)

(iv)Given:\((4m + 5n)^2 + (5m + 4n)^2\)

\(=[(4m)^2+2(4m)(5n)+(5n)^2] + [(5m)^2+2(5m)(4n)+(4n)^2]\)

\(=[16m^2+40mn+25n^2]+[25m^2+40mn+16n^2]\)

\(=(16m^2+25m^2)+(40mn+40mn)+(25n^2+16n^2)\quad\)[Placing same terms together]

\(=41m^2+80mn+41n^2\)

(v)Given: \((2.5p – 1.5q)^2 – (1.5p – 2.5q)^2\)

\(=[(2.5p)^2-2(2.5p)(1.5q)+(1.5q)^2] - [(1.5p)^2-2(1.5p)(2.5q)+(2.5q)^2]\)

\(=[6.25p^2-7.5pq+2.25q^2]-[2.25p^2-7.5pq+6.25q^2]\)

\(=6.25p^2-7.5pq+2.25q^2-2.25p^2+7.5pq-6.25q^2\)

\(=(6.25p^2-2.25p^2)+(-7.5pq+7.5pq)+(2.25q^2-6.25q^2)\quad\)[Placing same terms together]

\(=4p^2-4q^2\)

(vi)Given:\((ab + bc)^2 – 2ab^2c\)

\(=(ab)^2+2(ab)(bc)+(bc)^2-2ab^2c\)

\(=a^2b^2+2ab^2c+b^2c^2-2ab^2c\)

\(=a^2b^2+b^2c^2\)

(vii)Given:\((m^2 – n^2m)^2 + 2m^3n^2\)
\(=(m^2)^2-2(m^2)(n^2m)+(n^2m)^2+2m^3n^2\)

\(=m^4-2m^3n^2+n^4m^2+2m^3n^2\)

\(=m^4+n^4m^2\)