Question

разложить на множители
${x}^{4} + 8 {x}^{3} + 16 {x}^{2} - 25$
${x}^{2} + 6x - {y}^{2} + 4y + 5$
$(x + 3y)(x + 3y - 6) - (y + 3)(y - 3)$
$3 {x}^{2} - 4xy + {y}^{2} + 4x - 4$

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Answer 1

$1)~ x^4+8x^3+16x^2-25=(x^2+4x)^2-25=(x^2+4x-5)(x^2+4x+5)=\\ \\ =(x^2-x+5x-5)(x^2+4x+5)=(x(x-1)+5(x-1))(x^2+4x+5)=\\ \\ =(x-1)(x+5)(x^2+4x+5)$

$2)~ x^2+6x-y^2+4y+5=x^2+6x+9-(y^2-4y+4)=\\ \\ =(x+3)^2-(y-2)^2=(x+3-y+2)(x+3+y-2)=(x-y+5)(x+y+1)$

$3)~(x+3y)(x+3y-6)-(y+3)(y-3)=x^2+3xy-6x+3xy+9y^2-\\ -18y-y^2+9=x^2+6xy+8y^2-6x-18y+9=\\ =x^2+2xy+4xy+8y^2-3x-3x-12y-6y+9=\\ =x(x+2y-3)+4y(x+2y-3)-3(x+2y-3)=(x+2y-3)(x+4y-3)$

$4)~ 3x^2-4xy+y^2+4x-4=4x^2-4xy+y^2-x^2+4x-4=\\ \\ =(2x-y)^2-(x-2)^2=(2x-y-x+2)(2x-y+x-2)=(x-y+2)(3x-y-2)$