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Lời giải:
ĐKXĐ: $x\geq 0; x\neq 1; x\neq 25$

a) 

\(A=\frac{4\sqrt{x}}{\sqrt{x}-5}:\left[\frac{(\sqrt{x}-2)(\sqrt{x}+2)+\sqrt{x}-1}{(\sqrt{x}-1)(\sqrt{x}+2}+\frac{5-2\sqrt{x}}{(\sqrt{x}-1)(\sqrt{x}+2)}\right]\)

\(=\frac{4\sqrt{x}}{\sqrt{x}-5}:\frac{x-4+\sqrt{x}-1+5-2\sqrt{x}}{(\sqrt{x}-1)(\sqrt{x}+2)}\)

\(=\frac{4\sqrt{x}}{\sqrt{x}-5}:\frac{\sqrt{x}(\sqrt{x}-1)}{(\sqrt{x}-1)(\sqrt{x}+2)}=\frac{4\sqrt{x}}{\sqrt{x}-5}:\frac{\sqrt{x}}{\sqrt{x}+2}=\frac{4\sqrt{x}}{\sqrt{x}-5}.\frac{\sqrt{x}+2}{\sqrt{x}}=\frac{4(\sqrt{x}+2)}{\sqrt{x}-5}\)

b) Tại $x=81$ thì $\sqrt{x}=9$.

Khi đó: $A=\frac{4(9+2)}{9-5}=11$

c) $A< 4\Leftrightarrow \frac{\sqrt{x}+2}{\sqrt{x}-5}< 1$

$\Leftrightarrow \frac{7}{\sqrt{x}-5}< 0\Leftrightarrow \sqrt{x}-5< 0$

$\Leftrightarrow 0\leq x< 25$. Kết hợp với ĐKXĐ suy ra: $0\leq x< 25; x\neq 1$

a) A = (x - 5)(x² + 5x + 25) - (x - 2)(x + 2) + x(x² + x + 4)

= x³ - 125 - x² + 4 + x³ + x² + 4x

= (x³ + x³) + (-x² + x²) + 4x + (-125 + 4)

= 2x³ + 4x - 121

b) Tại x = -2 ta có:

A = 2.(-2)³ + 4.(-2) - 121

= 2.(-8) - 8 - 121

= -16 - 129

= -145

c) x² - 1 = 0

x² = 1

x = -1; x = 1

*) Tại x = -1 ta có:

A = 2.(-1)³ + 4.(-1) - 121

= 2.(-1) - 4 - 121

= -2 - 125

= -127

*) Tại x = 1 ta có:

A = 2.1³ + 4.1 - 121

= 2.1 + 4 - 121

= 2 - 117

= -115

\(x\ge-\dfrac{4}{5}\)

\(A=5x+2+\left|5x+4\right|=5x+2+5x+4=10x+6\)

\(x< -\dfrac{4}{5}\)

\(A=5x+2+\left|5x+4\right|=5x+2-5x+4=6\)

Với `x \ne -5,x \ne -1` có:

`A=[x+2]/[x+5]+[-5x-1]/[x^2+6x+5]-1/[1+x]`

`A=[(x+2)(x+1)-5x-1-(x+5)]/[(x+5)(x+1)]`

`A=[x^2+x+2x+2-5x-1-x-5]/[(x+5)(x+1)]`

`A=[x^2-3x-4]/[(x+5)(x+1)]`

`A=[(x-4)(x+1)]/[(x+5)(x+1)]`

`A=[x-4]/[x+5]`

\(=\dfrac{x+2}{x+5}+\dfrac{-5x-1}{x^2+x+5x+5}-\dfrac{1}{x+1}\\ =\dfrac{x+2}{x+5}+\dfrac{-5x-1}{\left(x^2+x\right)+\left(5x+5\right)}-\dfrac{1}{x+1}\\ =\dfrac{\left(x+2\right)\left(x+1\right)}{\left(x+1\right)\left(x+5\right)}+\dfrac{-5x-1}{x\left(x+1\right)+5\left(x+1\right)}-\dfrac{x+5}{\left(x+1\right)\left(x+5\right)}\\ =\dfrac{\left(x+2\right)\left(x+1\right)}{\left(x+1\right)\left(x+5\right)}+\dfrac{-5x-1}{\left(x+1\right)\left(x+5\right)}-\dfrac{x+5}{\left(x+1\right)\left(x+5\right)}\\ =\dfrac{x^2+2x+x+2-5x-1-x-5}{\left(x+1\right)\left(x+5\right)}\\ =\dfrac{x^2-3x-4}{\left(x+1\right)\left(x+5\right)}\\ =\dfrac{x^2+x-4x-4}{\left(x+1\right)\left(x+5\right)}\\ =\dfrac{\left(x^2+x\right)-\left(4x+4\right)}{\left(x+1\right)\left(x+5\right)}\\ =\dfrac{x\left(x+1\right)-4\left(x+1\right)}{\left(x+1\right)\left(x+5\right)}\\ =\dfrac{\left(x+1\right)\left(x-4\right)}{\left(x+1\right)\left(x+5\right)}\\ =\dfrac{x-4}{x+5}\)

ĐKXĐ: \(\left\{{}\begin{matrix}x\ge0\\x\ne4\end{matrix}\right.\)

Ta có: \(\dfrac{B}{A}=\dfrac{\sqrt{x}}{\sqrt{x}-2}:\dfrac{\sqrt{x}+5}{2\sqrt{x}-4}\)

\(\Leftrightarrow\dfrac{B}{A}=\dfrac{\sqrt{x}}{\sqrt{x}-2}\cdot\dfrac{2\left(\sqrt{x}-2\right)}{\sqrt{x}+5}\)

\(\Leftrightarrow\dfrac{B}{A}=\dfrac{2\sqrt{x}}{\sqrt{x}+5}\)

Để \(\dfrac{B}{A}\) nguyên thì \(2\sqrt{x}⋮\sqrt{x}+5\)

\(\Leftrightarrow2\sqrt{x}+10-10⋮\sqrt{x}+5\)

mà \(2\sqrt{x}+10⋮\sqrt{x}+5\)

nên \(-10⋮\sqrt{x}+5\)

\(\Leftrightarrow\sqrt{x}+5\inƯ\left(-10\right)\)

\(\Leftrightarrow\sqrt{x}+5\in\left\{1;-1;2;-2;5;-5;10;-10\right\}\)

\(\Leftrightarrow\sqrt{x}+5\in\left\{5;10\right\}\)(Vì \(\sqrt{x}+5\ge5\forall x\) thỏa mãn ĐKXĐ)

\(\Leftrightarrow\sqrt{x}\in\left\{0;5\right\}\)

hay \(x\in\left\{0;25\right\}\)

Kết hợp ĐKXĐ, ta được: \(x\in\left\{0;25\right\}\)

Vậy: Để \(\dfrac{B}{A}\) nguyên thì \(x\in\left\{0;25\right\}\)

a)Vì |4x - 2| = 6 <=> 4x - 2 ϵ {6,-6} <=> x ϵ {2,-1}

Thay x = 2, ta có B không tồn tại

Thay x = -1, ta có B = \(\dfrac{1}{3}\)

b)ĐKXĐ:x ≠ 2,-2

Ta có \(A=\dfrac{5}{x+2}+\dfrac{3}{2-x}-\dfrac{15-x}{4-x^2}=\dfrac{10-5x+3x+6}{\left(x+2\right)\left(2-x\right)}-\dfrac{15-x}{4-x^2}=\dfrac{16-2x}{\left(x+2\right)\left(2-x\right)}-\dfrac{15-x}{4-x^2}=\dfrac{2x-16}{\left(x+2\right)\left(x-2\right)}-\dfrac{15-x}{4-x^2}=\dfrac{2x-16}{x^2-4}+\dfrac{15-x}{x^2-4}=\dfrac{x-1}{x^2-4}\)c)Từ câu b, ta có \(A=\dfrac{x-1}{x^2-4}\)\(\Rightarrow\dfrac{2A}{B}=\dfrac{\dfrac{\dfrac{2x-2}{x^2-4}}{2x+1}}{x^2-4}=\dfrac{2x-2}{2x+1}< 1\) với mọi x

Do đó không tồn tại x thỏa mãn đề bài