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

a) \(xy-y\sqrt{x}+\sqrt{x}-1\)

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

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

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

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

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

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

a) \(\sqrt{a^3}-\sqrt{b^3}+\sqrt{a^2b}-\sqrt{ab^2}\)

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

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

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

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

b) \(x-y+\sqrt{xy^2}-\sqrt{y^3}\)

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

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

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

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

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

\(\text{b) }\sqrt{ax}-\sqrt{by}+\sqrt{bx}-\sqrt{xy}\text{ không phân tích được.}\)

\(\text{c) }=\left(\sqrt{x}-\sqrt{y}\right)\left(x+\sqrt{xy}+y\right)+\left(\sqrt{x}-\sqrt{y}\right).\sqrt{xy}\)

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

\(\text{d) }a+5\sqrt{a}+4=\sqrt{a}.\sqrt{a}+\sqrt{a}+4\sqrt{a}+4=\sqrt{a}\left(\sqrt{a}+1\right)+4\left(\sqrt{a}+1\right)\)

\(=\left(\sqrt{a}+1\right)\left(\sqrt{a}+4\right)\)

Bài 2: 

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: \(\sqrt{a-b}-\sqrt{a^2-b^2}\)

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

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