\(a\sqrt{a}+2a+\sqrt{a}+2=\left(a\sqrt{a}+2a\right)+\left(\sqrt{a}+2\right)\)

\(=a\left(\sqrt{a}+2\right)+\left(\sqrt{a}+2\right)=\left(\sqrt{a}+2\right)\left(a+1\right)\)

\(a\sqrt{a}+2a+\sqrt{a}+2=a\left(\sqrt{a}+2\right)+\left(\sqrt{a}+2\right)=\left(a+1\right)\left(\sqrt{a}+2\right)\)

\(a\sqrt{a}+2a+\sqrt{a}+2=\left(\sqrt{a}+2\right)\left(a+1\right)\)

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

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

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

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

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

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

\(a\sqrt{b}+\sqrt{ab}+\sqrt{a}+1\)

\(=\sqrt{ab}\cdot\sqrt{a}+\sqrt{ab}+\sqrt{a}+1\)

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

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

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

a√b + √(ab) + √a + 1

= [a√b + √(ab)] + (√a + 1)

= √(ab)(√a + 1) + (√a + 1)

= (√a + 1)[√(ab) + 1]

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

a)\(a-5\sqrt{a}=\sqrt{a}\left(\sqrt{a}-5\right)\)

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

c)\(a+4\sqrt{a}+4=\left(\sqrt{a}+2\right)^2\)

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