\(a^2+6a-(a+2)^2=a^2+6a-(a^2+4a+4)=2a-4\).

When \(a=-3\): \(2\cdot(-3)-4=-6-4=-10\).

\(\dfrac{2a^2-6a}{a-3}=\dfrac{2a(a-3)}{a-3}=2a\) (for \(a\ne 3\)).

So the expression becomes \(2a-4a+8=-2a+8\).

When \(a=4\): \(-2\cdot4+8=0\).

Factor out \((a-3)\): \((a-3)^2-a(a-3)=(a-3)\big((a-3)-a\big)=(a-3)(-3)=-3a+9\).

When \(a=-4\): \(-3\cdot(-4)+9=12+9=21\).

\(\dfrac{2a^3-4a^2}{a^2}=2a-4\) (for \(a\ne 0\)).

So the expression becomes \(2a-4-7a+3=-5a-1\).

When \(a=3\): \(-5\cdot3-1=-16\).

\((a-4)(a-3)=a^2-7a+12\).

Then \(a^2-7a+12-a^2+2a=-5a+12\).

When \(a=-2\): \(-5\cdot(-2)+12=10+12=22\).

Combine the fractions: \(\dfrac{a^2}{a-2}-\dfrac{4}{a-2}=\dfrac{a^2-4}{a-2}=\dfrac{(a-2)(a+2)}{a-2}=a+2\) (for \(a\ne2\)).

So the expression becomes \(a+2-5a+2=-4a+4\).

When \(a=3\): \(-12+4=-8\).

\((a-5)(a+5)=a^2-25\), and \(a(a-4)=a^2-4a\).

So \((a^2-25)-(a^2-4a)=4a-25\).

When \(a=-5\): \(4\cdot(-5)-25=-20-25=-45\).

\(3a+3=3(a+1)\).

Then \(\dfrac{a^2}{a+1}\cdot\dfrac{3(a+1)}{a}=3a\) (for \(a\ne0,-1\)).

The expression becomes \(3a-a-8=2a-8\).

When \(a=3\): \(6-8=-2\).

\((a+3)^2=a^2+6a+9\).

Then \(a^2+4a-(a^2+6a+9)=-2a-9\).

When \(a=-2\): \(-2\cdot(-2)-9=4-9=-5\).

\(2a^2+8a=2a(a+4)\), so \(\dfrac{2a^2+8a}{a+4}=2a\) (for \(a\ne-4\)).

The expression becomes \(2a-5a+2=-3a+2\).

When \(a=3\): \(-9+2=-7\).

\((a-4)^2=a^2-8a+16\), and \(a(a-6)=a^2-6a\).

So \((a^2-8a+16)-(a^2-6a)=-2a+16\).

When \(a=-3\): \(-2\cdot(-3)+16=6+16=22\).

\(\dfrac{2a^3+10a^2}{a^2}=2a+10\) (for \(a\ne0\)).

The expression becomes \(2a+10-5a+4=14-3a\).

When \(a=5\): \(14-15=-1\).

\((a-5)(a+3)=a^2-2a-15\).

Then \(a^2-2a-15-a^2+4a=2a-15\).

When \(a=-2\): \(2\cdot(-2)-15=-4-15=-19\).

\(\dfrac{a^2}{a+3}-\dfrac{9}{a+3}=\dfrac{a^2-9}{a+3}=\dfrac{(a-3)(a+3)}{a+3}=a-3\) (for \(a\ne-3\)).

So the expression becomes \(a-3-a+5=2\).

When \(a=10\), the value is \(2\).

\((a-4)(a+4)=a^2-16\), and \(a(a-6)=a^2-6a\).

So \((a^2-16)-(a^2-6a)=6a-16\).

When \(a=-3\): \(6\cdot(-3)-16=-18-16=-34\).

\(2a+10=2(a+5)\).

Then \(\dfrac{a^2}{a+5}\cdot\dfrac{2(a+5)}{a}=2a\) (for \(a\ne0,-5\)).

The expression becomes \(2a-4a+5=5-2a\).

When \(a=4\): \(5-8=-3\).