\(x^3-\dfrac{1}{8}\\ =x^3-\left(\dfrac{1}{2}\right)^3\\ =\left(x-\dfrac{1}{2}\right)\left(x^2+\dfrac{1}{2}x+\dfrac{1}{4}\right)\)

a: \(P=4\left(3^2+1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)\)

\(=\dfrac{\left(3^2-1\right)\left(3^2+1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)}{2}\)

\(=\dfrac{\left(3^4-1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)}{2}\)

\(=\dfrac{\left(3^8-1\right)\left(3^8+1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)}{2}\)

\(=\dfrac{\left(3^{16}-1\right)\left(3^{16}+1\right)\left(3^{32}+1\right)}{2}\)

\(=\dfrac{\left(3^{32}-1\right)\left(3^{32}+1\right)}{2}=\dfrac{3^{64}-1}{2}\)

b: \(Q=\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)\)

\(=\dfrac{\left(5^2-1\right)\left(5^2+1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)}{5^2-1}\)

\(=\dfrac{\left(5^4-1\right)\left(5^4+1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)}{5^2-1}\)

\(=\dfrac{\left(5^8-1\right)\left(5^8+1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)}{5^2-1}\)

\(=\dfrac{\left(5^{16}-1\right)\left(5^{16}+1\right)\left(5^{32}+1\right)}{5^2-1}\)

\(=\dfrac{\left(5^{32}-1\right)\left(5^{32}+1\right)}{24}=\dfrac{5^{64}-1}{24}\)

\(a.\left(x^2+5x+6\right)\left(x^2-15x+56\right)-144\\ =\left(x+2\right)\left(x+3\right)\left(x-7\right)\left(x-8\right)-144\\ =\left[\left(x+2\right)\left(x-7\right)\right]\left[\left(x+3\right)\left(x-8\right)\right]-144\\ =\left(x^2-5x-14\right)\left(x^2-5x-24\right)-144\\ =\left(x^2-5x-19+5\right)\left(x^2-5x-19-5\right)-144\\ =\left(x^2-5x-19\right)^2-5^2-144\\ =\left(x^2-5x-19\right)^2-169\\ =\left(x^2-5x+19\right)^2-13^2\\ =\left(x^2-5x-19-13\right)\left(x^2-5x-19+13\right)\\ =\left(x^2-5x-32\right)\left(x^2-5x-6\right)\\ =\left(x^2-5x-32\right)\left(x+1\right)\left(x-6\right)\) 

\(b.\left(x^2-11x+28\right)\left(x^2-7x+10\right)-72\\ =\left(x-4\right)\left(x-7\right)\left(x-5\right)\left(x-2\right)-72\\ =\left[\left(x-4\right)\left(x-5\right)\right]\left[\left(x-7\right)\left(x-2\right)\right]-72\\ =\left(x^2-9x+20\right)\left(x^2-9x+14\right)-72\\ =\left(x^2-9x+17+3\right)\left(x^2-9x+17-3\right)-72\\ =\left(x^2-9x+17\right)^2-3^2-72\\ =\left(x^2-9x+17\right)^2-81\\ =\left(x^2-9x+17\right)^2-9^2\\ =\left(x^2-9x+17-9\right)\left(x^2-9x+17+9\right)\\ =\left(x^2-9x+8\right)\left(x^2-9x+26\right)\\ =\left(x-1\right)\left(x-8\right)\left(x^2-9x+26\right)\)

a)

A = -4x² - 12x

= -4(x² + 3x)

b)

B = 3 - 4x - x²

= -(x² + 4x - 3)

= -(x² + 4x + 4 - 7)

= -(x + 2)² + 7

Do (x + 2)² ≥ 0

⇒ -(x + 2)² ≤ 0

⇒ -(x + 2)² + 7 ≤ 7

Vậy maxB = 7 khi x = -2

\(x^{64}+x^{32}+1\\ =x^{64}+2x^{32}+1+x^{32}-2x^{32}\\ =\left[\left(x^{32}\right)^2+2\cdot x^{32}\cdot1+1^2\right]-x^{32}\\ =\left(x^{32}+1\right)^2-\left(x^{16}\right)^2\\ =\left(x^{32}-x^{16}+1\right)\left(x^{32}+x^{16}+1\right)\\ =\left(x^{32}-x^{16}+1\right)\left[\left(x^{32}+2x^{16}+1\right)+x^{16}-2x^{16}\right]\\ =\left(x^{32}-x^{16}+1\right)\left[\left(x^{16}+1\right)^2-x^{16}\right]\\ =\left(x^{32}-x^{16}+1\right)\left(x^{16}-x^8+1\right)\left(x^{16}+x^8+1\right)\\ =\left(x^{32}-x^{16}+1\right)\left(x^{16}-x^8+1\right)\left[\left(x^{16}+2x^8+1\right)-x^8\right]\\ =\left(x^{32}-x^{16}+1\right)\left(x^{16}-x^8+1\right)\left[\left(x^8+1\right)^2-x^8\right]\\ =\left(x^{32}-x^{16}+1\right)\left(x^{16}-x^8+1\right)\left(x^8-x^4+1\right)\left(x^8+x^4+1\right)\\ =\left(x^{32}-x^{16}+1\right)\left(x^{16}-x^8+1\right)\left(x^8-x^4+1\right)\left[\left(x^8+2x^4+1\right)-x^4\right]\\ =\left(x^{32}-x^{16}+1\right)\left(x^{16}-x^8+1\right)\left(x^8-x^4+1\right)\left[\left(x^4+1\right)^2-x^4\right]\\ =\left(x^{32}-x^{16}+1\right)\left(x^{16}-x^8+1\right)\left(x^8-x^4+1\right)\left(x^4-x^2+1\right)\left(x^4+x^2+1\right)\)

\(=\left(x^{32}-x^{16}+1\right)\left(x^{16}-x^8+1\right)\left(x^8-x^4+1\right)\left(x^4-x^2+1\right)\left[\left(x^4+2x^2+1\right)-x^2\right]\\ =\left(x^{32}-x^{16}+1\right)\left(x^{16}-x^8+1\right)\left(x^8-x^4+1\right)\left(x^4-x^2+1\right)\left[\left(x^2+1\right)^2-x^2\right]\\ =\left(x^{32}-x^{16}+1\right)\left(x^{16}-x^8+1\right)\left(x^8-x^4+1\right)\left(x^4-x^2+1\right)\left(x^2-x+1\right)\left(x^2+x+1\right)\)

\(x^{64}+x^{32}+1\)

\(=x^{64}+2x^{32}-x^{32}+1\)

\(=\left(x^{64}+2^{32}+1\right)-x^{32}\)

\(=\left(x^{32}+1\right)^2-\left(x^{16}\right)^2\)

\(=\left(x^{32}+1-x^{16}\right)\left(x^{32}+1+x^{16}\right)\)

\(\left(x^2-4\right)\left(x^2-10\right)-72\)

\(=\left(x^2-7+3\right)\left(x^2-7-3\right)-72\)

\(=\left(x^2-7\right)^2-9-72\)

\(=\left(x^2-7\right)^2-81\)

\(=\left(x^2-7+9\right)\left(x^2-7-9\right)\)

\(=\left(x^2+2\right)\left(x^2-16\right)\)

\(=\left(x-4\right)\left(x+4\right)\left(x^2+2\right)\)

\(\left(x^2-4\right)\left(x^2-10\right)-72\)

\(=x^4-10x^2-4x^2+40-72\)

\(=x^4-14x^2-32\)

\(=\left(x^2-16\right)\left(x^2+2\right)=\left(x-4\right)\left(x+4\right)\left(x^2+2\right)\)

\(x^4+6x^3+7x^2-6x+1\\ =\left(x^4+3x^3-x^2\right)+\left(3x^3+9x^2-x\right)-\left(x^2+3x-1\right)\\ =x^2\left(x^2+3x-1\right)+3x\left(x^2+3x-1\right)-\left(x^2+3x-1\right)\\ =\left(x^2+3x-1\right)\left(x^2+3x-1\right)\\ =\left(x^2+3x-1\right)^2\)

`= x^2(x^2 + x + 1) - x(x^2 + x + 1) + 2024(x^2 + x + 1)`

`= (x^2 - x + 2024)(x^2 + x + 1)`.

\(3\cdot\left(2\cdot1\right)=6x-3x\\ 6=3x\\ x=\dfrac{6}{3}=2\)