a) \(\left(3x+5\right)^2=\left(3x\right)^2+2.3x.5+5^2=9x^2+30x+25\)
b) \(\left(6x^2+\dfrac{1}{3}\right)^2=\left(6x^2\right)^2+2.6x^2.\dfrac{1}{3}+\left(\dfrac{1}{3}\right)^2=36x^4+4x^2+\dfrac{1}{9}\)
c) \(\left(5x-4y\right)^2=\left(5x\right)^2-2.5x.4y+\left(4y\right)^2=25x^2-40xy+16y^2\)
d) \(\left(2x^2y-3y^3x\right)^2=\left(2x^2y\right)^2-2.2x^2y.3y^3x+\left(3y^3x\right)^2=4x^4y-12x^3y^4+9x^2y^6\)
e) \(\left(5x-3\right)\left(5x+3\right)=\left(5x\right)^2-3^2=25x^2-9\)
f) \(\left(6x+5y\right)\left(6x-5y\right)=\left(6x\right)^2-\left(5y\right)^2=36x^2-25y^2\)
g) \(\left(-4xy-5\right)\left(5-4xy\right)=-\left(5+4xy\right)\left(5-4xy\right)\)
\(=-\left[5^2-\left(4xy\right)^2\right]=-\left(25-16x^2y^2\right)=16x^2y^2-25\)
h) \(\left(a^2b+ab^2\right)\left(ab^2-a^2b\right)=\left(ab^2\right)^2-\left(a^2b\right)^2=a^2b^4-a^4b^2\)
i) \(\left(3x-4\right)^2+2.\left(3x-4\right)\left(4-x\right)+\left(4-x\right)^2\)
\(=\left(3x-4+4-x\right)^2=\left(2x\right)^2=4x^2\)
j) \(\left(3a-1\right)^2-2\left(9a^2-1\right)+\left(3a+1\right)^2\)
\(=\left(3a-1\right)^2-2\left(3a-1\right)\left(3a+1\right)+\left(3a+1\right)^2\)
\(=\left(3a-1-3a-1\right)^2=\left(-2\right)^2=4\)
k) \(\left(a^2+ab+b^2\right)\left(a^2-ab+b^2\right)-\left(a^4+b^4\right)\)
\(=\left(a^2+b^2\right)^2-\left(ab\right)^2-a^4-b^4\)
\(=a^4+2a^2b^2+b^4-a^2b^2-a^4-b^4\)
\(=a^2b^2\)
