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\(-25x^2\sqrt{2}+10x+4\sqrt{2}=-\sqrt{2}\left(25x^2-\dfrac{10}{\sqrt{2}}-4\right)=-\sqrt{2}.\left(\left(25x\right)^2-2.5.\dfrac{1}{\sqrt{2}}+\dfrac{1}{2}-\dfrac{5}{2}\right)=-\sqrt{2}\left[\left(5x-\dfrac{1}{\sqrt{2}}\right)^2-\dfrac{5}{2}\right]=-\sqrt{2}.\left(5x-\dfrac{1}{\sqrt{2}}-\dfrac{\sqrt{5}}{\sqrt{2}}\right).\left(5x-\dfrac{1}{\sqrt{2}}+\dfrac{\sqrt{5}}{\sqrt{2}}\right)=-\sqrt{2}.\left(5x-\dfrac{1+\sqrt{5}}{\sqrt{2}}\right)\left(5x-\dfrac{1-\sqrt{5}}{\sqrt{2}}\right)\)
\(\left(25x^2-2\right)=\left(5x-\sqrt[]{2}\right)\left(5x+\sqrt[]{2}\right)\)
\(\text{e) 2x^4 + 5x^3+13x^2+25x+15 }\)
\(\text{=2x^3(x+1)+3x^2(x+1)+10x(x+1)+15(x+1) }\)
\(\text{=(x+1)[x^2(2x+3)+5(2x+3)]}\)
\(\text{=(x+1)(2x+3)(x^2+5)}\)
\(2x^4+5x^3+13x^2+25x+15\)
\(=2x^4+2x^3+3x^3+3x^2+10x^2+10x+15x+15\)
\(=2x^3\left(x+1\right)+3x^2\left(x+1\right)+10x\left(x+1\right)+15\left(x+1\right)\)
\(=\left(x+1\right)\left(2x^3+3x^2+10x+15\right)\)
\(25x^2-70xy+49y^2\\ =25x^2-35xy-35xy+49y^2\\ =25x\left(x-\dfrac{7}{5}y\right)-35y\left(x-\dfrac{7}{5}y\right)\\ =\left(25x-35y\right)\left(x-\dfrac{7}{5}y\right)\\ =5\left(5x-7y\right)\left(x-\dfrac{7}{5}y\right)\)
\(25x^2-4y^2-4y-1\)
\(=25x^2-\left(2y+1\right)^2=\left(5x-2y-1\right)\left(5x+2y+1\right)\)
25x2 - 4y2 - 4y - 1
= 25x2 - (4y2 + 4y + 1)
= (5x)2 - (2y + 1)2
= [5x - (2y + 1)][5x + (2y + 1)]
= (5x - 2y - 1)(5x + 2y + 1)