1. The problem is to analyze the polynomial expression $3x + 12$. 2. This is a linear polynomial of the form $ax + b$ where $a=3$ and $b=12$. 3. To find the roots, set the polynomial equal to zero: $$3x + 12 = 0$$ 4. Solve for $x$: $$3x = -12$$ $$\cancel{3}x = \cancel{3}(-4)$$ $$x = -4$$ 5. The root of the polynomial is $x = -4$. 6. The graph of this polynomial is a straight line with slope 3 and y-intercept 12. 7. The y-intercept is the point where $x=0$, so $y=3(0)+12=12$. 8. The x-intercept is the root found above, $x=-4$. Final answer: The polynomial $3x + 12$ has a root at $x = -4$ and y-intercept at $y=12$.