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## Graphing Quadratic Inequalities 1.1 - Solution

1. Graph the related quadratic function.
2. Test a point not on the parabola.
3. Shade accordingly. If the point satisfies the inequality, we shade the region that contains the point. If not, we shade the opposite region.

### Step $1$

Let's draw the graph of the related function, which is $y=x^2+4x+3.$

### Step $2$

Next, let's determine which region to shade by testing a point. For simplicity, we will use $(0,0)$ as our test point. Let's see if it satisfies the given inequality.
$y\leq x^2+4x+3$
${\color{#009600}{0}}\stackrel{?}{\leq} ({\color{#0000FF}{0}})^2+4({\color{#0000FF}{0}})+3$
$0\stackrel{?}{\leq} 0+4(0)+3$
$0\stackrel{?}{\leq} 0+0+3$
$0\leq 3 \Large{\color{#009600}{ \checkmark}}$

### Step $3$

Since $(0,0)$ produced a true statement, we will shade the region that contains the point. Also, note that the inequality is not strict. Therefore, the parabola will be solid.

This graph corresponds to option III.