By choosing points not on the logo and Substituting them into the inequality of my logo, show why the inequality used to create the shading give the region created \[ \text { Inequality }=(2 x+45)^{2}+(2 y-0)^{2} \leqslant 5^{2} \]

Step 1 ：First, we need to understand the inequality. The inequality $(2x+45)^2+(2y-0)^2\leq 25$ represents a circle with center $(-22.5, 0)$ and radius 5.

Step 2 ：Let's choose a point not on the logo, for example, the origin $(0,0)$. We will substitute this point into the inequality to see if it satisfies the inequality.

Step 3 ：Substitute $(0,0)$ into the inequality: $(2(0)+45)^2+(2(0)-0)^2\leq 25$

Step 4 ：Simplify the expression: $(45)^2+(0)^2\leq 25$

Step 5 ：Calculate the result: $2025\leq 25$, which is false.

Step 6 ：Since the point $(0,0)$ does not satisfy the inequality, it is not in the shaded region.

Step 7 ：Now, let's choose a point inside the logo, for example, $(-22,0)$. We will substitute this point into the inequality to see if it satisfies the inequality.

Step 8 ：Substitute $(-22,0)$ into the inequality: $(2(-22)+45)^2+(2(0)-0)^2\leq 25$

Step 9 ：Simplify the expression: $(1)^2+(0)^2\leq 25$

Step 10 ：Calculate the result: $1\leq 25$, which is true.

Step 11 ：Since the point $(-22,0)$ satisfies the inequality, it is in the shaded region.

Step 12 ：\(\boxed{\text{The inequality correctly represents the shaded region.}}\)