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a) (a² - 1)² + 4a²

= a⁴ - 2a² + 1 + 4a²

= (a²)² + 2.a².1 + 1²)

=(a² + 1)²

b) (6x² + y²)(y² - 6x²)

= (y² + 6x²)(y² - 6x²)

= (y²)² - (6x²)²

= y⁴ - 36x⁴

\(A+B=\left(5x^2-3xy+7x^2\right)+\left(6x^2-8xy+9y^2\right)\)

\(A+B=5x^2-3xy+7x^2+6x^2-8xy+9y^2\)

\(A+B=\left(5x^2+7x^2+6x^2+9x^2\right)+\left(-3xy-8xy\right)\)

\(A+B=27x^2+\left(-11xy\right)\)

\(16x^2-8xy+y^2+1=\left(4x-y\right)^2+1\ge1\)

Dấu \("="\Leftrightarrow4x=y\)

\(-4x^2+2x-1=-\left(4x^2-2\cdot2\cdot\dfrac{1}{2}x+\dfrac{1}{16}\right)-\dfrac{15}{16}=-\left(2x-\dfrac{1}{4}\right)^2-\dfrac{15}{16}\le-\dfrac{15}{16}\)

Dấu \("="\Leftrightarrow2x=\dfrac{1}{4}\Leftrightarrow x=\dfrac{1}{8}\)

Bài 12:

a) \(\left(\dfrac{1}{2}x+4\right)^2\)

\(=\left(\dfrac{1}{2}x\right)^2+2\cdot\dfrac{1}{2}x\cdot4+4^2\)

\(=\dfrac{1}{4}x^2+4x+16\)

b) \(\left(7x-5y\right)^2\)

\(=\left(7x\right)^2-2\cdot7x\cdot5y+\left(5y\right)^2\)

\(=49x^2-70xy+25y^2\)

c) \(\left(6x^2+y^2\right)\left(y^2-6x^2\right)\)

\(=\left(y^2+6x^2\right)\left(y^2-6x^2\right)\)

\(=y^4-36x^4\)

d) \(\left(x+2y\right)^2\)

\(=x^2+2\cdot x\cdot2y+\left(2y\right)^2\)

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

e) \(\left(x-3y\right)\left(x+3y\right)\)

\(=x^2-\left(3y\right)^2\)

\(=x^2-9y^2\)

f) \(\left(5-x\right)^2\)

\(=5^2-2\cdot5\cdot x+x^2\)

\(=25-10x+x^2\)

\(11,\)

\(a,\left(7x+4\right)^2-\left(7x+4\right)\left(7x-4\right)\)

\(=\left(7x+4\right)\left(7x+4-7x+4\right)\)

\(=\left(7x+4\right).8=56x+32\)

\(b,\left(x+2y\right)^2-6xy\left(x+2y\right)\)

\(=\left(x+2y\right)\left(x+2y-6xy\right)\)

a) \(=3xy\left(2x-3\right)\)

b) \(=\left(y+5\right)^2\)

\(a,=\left(6a^2-y\right)\left(6a^2+y\right)\\ b,=\left(x-2y\right)^2\\ c=\left(6x^2-6x\right)+\left(x-1\right)=6x\left(x-1\right)+\left(x-1\right)=\left(x-1\right)\left(6x+1\right)\)

f: \(\left(y-1\right)^2\)

e: =(x+y+5)(x+y-5)

\(d,15x^2y+20xy^2+25xy=5xy\left(3x+4y+5\right)\\ e,\left(x+y\right)^2-25=\left(x+y\right)^2-5^2=\left(x+y-5\right)\left(x+y+5\right)\\ f,1-2y+y^2=\left(1-y\right)^2\\ h,4x^2+8xy-3x-6y=\left(4x^2+8xy\right)-\left(3x+6y\right)=4x\left(x+2y\right)-3\left(x+2y\right)=\left(x+2y\right)\left(4x-3\right)\)