1: \(x^3+3x^2-x-3=\left(x-2\right)\left(x^2+bx+c\right)+a\)

\(\Leftrightarrow x^3-2x^2+5x^2-10x+11x-22+19=\left(x-2\right)\left(x^2+bx+c\right)+a\)

\(\Leftrightarrow\left(x-2\right)\left(x^2+5x+11\right)+19=\left(x-2\right)\left(x^2+bx+c\right)+a\)

=>b=5; c=11; c=19

2: \(4x^3+7x-6=\left(ax+b\right)\left(x^2+x+1\right)+c\)

\(\Leftrightarrow4x^3+4x^2+4x-4x^2-4x-4+7x-2=\left(ax+b\right)\left(x^2+x+1\right)+c\)

\(\Leftrightarrow\left(x^2+x+1\right)\left(4x-4\right)+7x-2=\left(ax+b\right)\left(x^2+x+1\right)+c\)

=>a=4; b=-4; c=7x-2

`@` `\text {Ans}`

`\downarrow`

`B(x)-A(x)+C(x)`

`=`\((x^2-5x^3-4x+7) - (-x^3 + 7x^2 +2x - 15) + 3x^3 - 7x^2 -4\)

`=`\(x^2-5x^3-4x+7+x^3-7x^2-2x+15+3x^3-7x^2-4\)

`=`\(\left(-5x^3+x^3+3x^3\right)+\left(x^2-7x^2-7x^2\right)+\left(-4x-2x\right)+\left(7+15-4\right)\)

`=`\(-x^3-13x^2-6x+18\)

`C(x)-B(x)-A(x)`

`=`\(3x^3 - 7x^2 -4 - (x^2-5x^3-4x+7) - (-x^3 + 7x^2 +2x - 15)\)

`=`\(3x^3-7x^2-4-x^2+5x^3+4x-7+x^3-7x^2-2x+15\)

`=`\(\left(3x^3+5x^3+x^3\right)+\left(-7x^2-x^2-7x^2\right)+\left(4x-2x\right)+\left(-4-7+15\right)\)

`=`\(9x^3-15x^2+2x+4\)

a) \(B\left(x\right)-A\left(x\right)+C\left(x\right)\)

\(=\left(x^2-5x^3-4x+7\right)-\left(-x^3+7x^2+2x-15\right)+\left(3x^3-7x^2-4\right)\)

\(=x^2-5x^3-4x+7+x^3-7x^2-2x+15+3x^3-7x^2-4\)

\(=\left(-5x^3+x^3+3x^3\right)+\left(x^2-7x^2-7x^2\right)-\left(4x+2x\right)+\left(7-4+15\right)\)

\(=-x^3-13x^2-6x+18\)

b) \(C\left(x\right)-B\left(x\right)-A\left(x\right)\)

\(=\left(3x^3-7x^2-4\right)-\left(x^2-5x^3-4x+7\right)-\left(-x^3+7x^2+2x-15\right)\)

\(=3x^3-7x^2-4-x^2+5x^3+4x-7+x^3-7x^2-2x+15\)

\(=\left(3x^3+5x^3+x^3\right)-\left(7x^2+x^2+7x^2\right)+\left(4x-2x\right)-\left(4+7-15\right)\)

\(=9x^3-15x^2+2x+4\)

\(B\left(x\right)-A\left(x\right)+C\left(x\right)\)

\(=\left(x^2-5x^3-4x+7\right)-\left(-x^3+7x^2+2x-15\right)+\left(3x^3-7x^2-4\right)\)

\(=x^2-5x^3-4x+7+x^3-7x^2-2x+15+3x^3-7x^2-4\)

\(=\left(-5x^3+x^3+3x^3\right)+\left(x^2-7x^2-7x^2\right)+\left(-4x-2x\right)+\left(7+15-4\right)\)

\(=-x^3-13x^2-6x+18\)

\(C\left(x\right)-B\left(x\right)-A\left(x\right)\)

\(=\left(3x^3-7x^2-4\right)-\left(x^2-5x^3-4x+7\right)-\left(-x^3+7x^2+2x-15\right)\)

\(=3x^3-7x^2-4-x^2+5x^3+4x-7+x^3-7x^2-2x+15\)

\(=\left(3x^3+5x^3+x^3\right)+\left(-7x^2-x^2-7x^2\right)+\left(4x-2x\right)+\left(-4-7+15\right)\)

\(=9x^3-15x^2+2x+4\)

b. 6x(x - 5) - x(6x + 3)

= x(6x - 30) - x(6x + 3)

= x(6x - 30 - 6x - 3)

= x(-33)

= -33x

\(1,\\ a,=-35x^5y^4z\\ b,=6x^2-30x-6x^2-3x=-33x\\ c,=x^3-9x^2-2x^2+18x-x+9=x^3-11x^2+17x+9\\ 2,\\ A\left(x\right)+B\left(x\right)=10-2x+4x^3-5x^2-10x^3-5x+6x^2-20\\ =-6x^3+x^2-7x-10\\ A\left(x\right)-B\left(x\right)=10-2x+4x^3-5x^2+10x^3+5x-6x^2+20\\ =14x^3-11x^2+3x+30\\ 3,\\ a,M\left(x\right)=5x+20=0\\ \Leftrightarrow x=-4\\ b,N\left(x\right)=100x^2-49=0\\ \Leftrightarrow\left(10x-7\right)\left(10x+7\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=\dfrac{7}{10}\\x=-\dfrac{7}{10}\end{matrix}\right.\\ c,P\left(x\right)=3x-15=0\\ \Leftrightarrow x=5\)

Bài 1;

a)\(5x^3yz.\left(-7x^2y^3\right)=-35.x^5y^4z\)

b)\(6x\left(x-5\right)-x\left(6x+3\right)=6x^2-30x-6x^2-3x=-33x\)

c) \(\left(x-9\right)\left(x^2-2x-1\right)=x^3-2x^2-x-9x^2+18x+9=x^3-11x^2+17x+9\)