d: \(=-\left(x+\sqrt{x}-12\right)=-\left(\sqrt{x}+4\right)\left(\sqrt{x}-3\right)\)

\(a)\) \(xy-y\sqrt{x}+\sqrt{x}-1\)

= \(y\sqrt{x}.(\sqrt{x}-1)+\sqrt{x}-1\)

=\((\sqrt{x}-1).(y\sqrt{x}+1)\).

\(b)\)\(\sqrt{ax}-\sqrt{by}+\sqrt{bx}-\sqrt{ay}\)

=\(\sqrt{a}.\sqrt{x}-\sqrt{b}.\sqrt{y}+\sqrt{b}.\sqrt{x}-\sqrt{a}.\sqrt{y}\)

=\(\sqrt{a}.\sqrt{x}+\sqrt{b}.\sqrt{x}-\sqrt{a}.\sqrt{y}-\sqrt{b}.\sqrt{y}\)

=\(\sqrt{x}.(\sqrt{a}+\sqrt{b})-\sqrt{y}.(\sqrt{a}+\sqrt{b})\)

=\((\sqrt{x}-\sqrt{y}).(\sqrt{a}+\sqrt{b})\).

\(c)\)\(\sqrt{a+b}+\sqrt{a^2-b^2}\)

=\(\sqrt{a+b}+\sqrt{(a+b).(a-b)}\)

=\(\sqrt{a+b}+\sqrt{a+b}.\sqrt{a-b}\)

=\(\sqrt{a+b}.\left(1+\sqrt{a-b}\right)\).

\(d)\) \(12-\sqrt{x}-x\)

=\(12-4\sqrt{x}+3\sqrt{x}-x\)

=\(4.\left(3-\sqrt{x}\right)+\sqrt{x}\left(3-\sqrt{x}\right)\)

=\(\left(3-\sqrt{x}\right).\left(4+\sqrt{3}\right)\).

a) \(xy-y\sqrt{x}+\sqrt{x}-1=\left(\sqrt{x}\right)^2.y-y\sqrt{x}+\sqrt{x}-1\)

\(=y\sqrt{x}\left(\sqrt{x}-1\right)+\left(\sqrt{x}-1\right)\)

\(=\left(\sqrt{x}-1\right)\left(y\sqrt{x}+1\right)\)

b) \(\sqrt{ax}-\sqrt{by}+\sqrt{bx}-\sqrt{ay}=\left(\sqrt{ax}+\sqrt{bx}\right)-\left(\sqrt{ay}+\sqrt{by}\right)\)

\(=\sqrt{x}\left(\sqrt{a}+\sqrt{b}\right)-\sqrt{y}\left(\sqrt{a}+\sqrt{b}\right)\)

\(=\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{x}-\sqrt{y}\right)\)

c) \(\sqrt{a+b}+\sqrt{a^2-b^2}=\sqrt{a+b}+\sqrt{\left(a-b\right)\left(a+b\right)}\)

\(=\sqrt{a+b}+\sqrt{a-b}.\sqrt{a+b}\)

\(=\sqrt{a+b}.\left(1+\sqrt{a-b}\right)\)

d) \(12-\sqrt{x}-x=12-\sqrt{4x}+\sqrt{3x}-x\)

\(=4\left(3-\sqrt{x}\right)+\sqrt{x}\left(3-\sqrt{x}\right)\)

\(=\left(3-\sqrt{x}\right)\left(4+\sqrt{x}\right)\)

a/ \(=\sqrt{a}\left(\sqrt{x}-\sqrt{y}\right)-\sqrt{b}\left(\sqrt{x}-\sqrt{y}\right)=\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{a}-\sqrt{b}\right)\)

b/ \(=\sqrt{\left(a-b\right)\left(a+b\right)}-\sqrt{\left(a+b\right)\left(a^2-ab+b^2\right)}\)

\(=\sqrt{a+b}\left(\sqrt{\left(a-b\right)\left(a^2-ab+b^2\right)}\right)\)

c/ \(=\left(\sqrt{x}-\frac{3}{2}\right)^2-\frac{81}{4}=\left(\sqrt{x}-\frac{3}{2}-\frac{9}{2}\right)\left(\sqrt{x}-\frac{3}{2}+\frac{9}{2}\right)=\left(\sqrt{x}-6\right)\left(\sqrt{x}+3\right)\)

\(a.\sqrt{ax}+\sqrt{by}-\sqrt{ay}-\sqrt{bx}\\ =\left(\sqrt{ax}-\sqrt{ay}\right)-\left(\sqrt{bx}-\sqrt{by}\right)\\ =\sqrt{a}\left(\sqrt{x}-\sqrt{y}\right)-\sqrt{b}\left(\sqrt{x}-\sqrt{y}\right)\\ =\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{x}-\sqrt{y}\right)\)

\(b.\sqrt{a^2-b^2}-\sqrt{a^3+b^3}\\ =\sqrt{\left(a+b\right)\left(a-b\right)}-\sqrt{\left(a+b\right)\left(a^2-ab+b^2\right)}\\ =\sqrt{a+b}\left(\sqrt{a-b}-\sqrt{a^2-ab+b^2}\right)\)

\(c.x-3\sqrt{x}-18=x-6\sqrt{x}+3\sqrt{x}-18\\ =\sqrt{x}\left(\sqrt{x}-6\right)+3\left(\sqrt{x}-6\right)\\ =\left(\sqrt{x}+3\right)\left(\sqrt{x}-6\right)\)

\(d.x\sqrt{x}+4x-12\sqrt{x}-27=\left(\sqrt{x^3}-27\right)+\left(4x-12\sqrt{x}\right)\\ =\left(\sqrt{x}-3\right)\left(x+3\sqrt{x}+9\right)+4\sqrt{x}\left(\sqrt{x}-3\right)\\ =\left(\sqrt{x}-3\right)\left(x+3\sqrt{x}+9+4\sqrt{x}\right)\\ =\left(\sqrt{x}-3\right)\left(x+7\sqrt{x}+9\right)\)

(có gì sai mong mọi người góp ý)

a, \(\sqrt{ax}-\sqrt{by}+\sqrt{bx}-\sqrt{ay}\)

\(=\sqrt{x}\left(\sqrt{a}+\sqrt{b}\right)-\sqrt{y}\left(\sqrt{a}+\sqrt{b}\right)\)

\(=\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{a}+\sqrt{b}\right)\)

b, \(x^2+x\sqrt{x}+\sqrt{x}+1\)

\(=\sqrt{x}\left(x\sqrt{x}+1\right)+\left(x\sqrt{x}+1\right)\)

\(=\left(\sqrt{x}+1\right)\left(x\sqrt{x}+1\right)\)

c, \(\sqrt{6x}-x2\sqrt{3}\)

\(=\sqrt{6x}-\sqrt{12x^2}\)

\(=\sqrt{6x}\left(1-\sqrt{2x}\right)\)

Câu a : \(\sqrt{xy}-x=\sqrt{x}\left(\sqrt{y}-\sqrt{x}\right)\)

Câu b : \(x+y-2\sqrt{xy}=\left(\sqrt{x}-\sqrt{y}\right)^2=\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)\)

Câu c : \(\sqrt{xy}+2\sqrt{x}-3\sqrt{y}-6=\sqrt{x}\left(\sqrt{y}+2\right)-3\left(\sqrt{y}+2\right)=\left(\sqrt{x}-3\right)\left(\sqrt{y}+2\right)\)

Câu d : \(x\sqrt{y}-y\sqrt{x}=\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)\)

a) \(\sqrt{ax}-\sqrt{by}+\sqrt{bx}-\sqrt{ay}\)

\(=\sqrt{a}\left(\sqrt{x}-\sqrt{y}\right)+\sqrt{b}\left(\sqrt{x}-\sqrt{y}\right)\)

\(=\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{a}+\sqrt{b}\right)\)

b) \(7\sqrt{ab}+7b-\sqrt{a}-\sqrt{b}\)

\(=7\sqrt{b}\left(\sqrt{a}+\sqrt{b}\right)-\left(\sqrt{a}+\sqrt{b}\right)\)

\(=\left(\sqrt{a}+\sqrt{b}\right)\left(7\sqrt{b}-1\right)\)

c) \(a\sqrt{b}-b\sqrt{a}+\sqrt{a}-\sqrt{b}\)

\(=\sqrt{a}\left(\sqrt{ab}+1\right)-\sqrt{b}\left(\sqrt{ab}+1\right)\)

\(=\left(\sqrt{ab}+1\right)\left(\sqrt{a}-\sqrt{b}\right)\)

d) \(\sqrt{x^2-25y^2}-\sqrt{x-5y}\)

\(=\sqrt{\left(x-5y\right)\left(x+5y\right)}-\sqrt{x-5y}\)

\(=\sqrt{x-5y}\left(\sqrt{x+5y}-1\right)\)