Answer
See below
Work Step by Step
Let $u=(u_1,u_2)\\
v=(v_1,v_2)\\
w=(w_1,w_2)$
1. Property:
$=v_1v_1-v_1v_2-v_2v_1+4v_2v_2\\
=v_1^2-2v_1v_2+4v_2^2\\
=v_1^2-2v_1v_2+v_2^2+3v_2^2\\
=(v_1-v_2)^2+3v_2^2 \geq 0$
If $v=0 \rightarrow =(0-0)^2+3.0=0$
If $v \ne 0 \rightarrow (v_1-v_2)^2+3v_2^2 \gt 0$
Hence, the property $=0 \leftrightarrow v=0$ does not hold.
2. Property
$=v_1w_1-v_1w_2-v_2w_1+4v_2w_2\\
=w_1v_1-w_1v_2-w_2v_1+4w_2v_2\\
=$
3. Property
$=(kv_1)w_1-(kv_1)w_2-(kv_2)w_1+4(kv_2)w_2\\
=kv_1w_1-kv_1w_2-kv_2w_1+4kv_2w_2\\
=k(v_1w_1-v_1w_2-v_2w_1+4v_2w_2)\\
=k$
4. Property
$=(u_1+v_1)w_1-(u_1+v_1)w_2-(u_2+v_2)w_1+4(u_2+v_2)w_2\\
=u_1w_1+v_1w_1-u_1w_2-v_1w_2-u_2w_1-v_2w_1+4u_2w_2+4v_2w_2\\
=(u_1w_1-u_1w_2-u_2w_1+4u_2w_2)+(v_1w_1-v_1w_2-v_2w_1+4v_2w_2)\\
=+$
