1. **State the problem:** We need to determine which polynomial function passes through the points (2,5), (3,2), and (4,5). 2. **Given functions:** - $p(x)=19-18x+3x^2$ - $p(x)=29-18x+3x^2$ - $p(x)=19-18x+4x^2$ - $p(x)=29-18x+4x^2$ 3. **Approach:** For each polynomial, substitute $x=2$, $x=3$, and $x=4$ and check if the output equals the corresponding $y$ values 5, 2, and 5. 4. **Check $p(x)=19-18x+3x^2$:** - $p(2)=19-18(2)+3(2)^2=19-36+12=-5$ (should be 5, so no) 5. **Check $p(x)=29-18x+3x^2$:** - $p(2)=29-36+12=5$ (matches) - $p(3)=29-54+27=2$ (matches) - $p(4)=29-72+48=5$ (matches) 6. **Check $p(x)=19-18x+4x^2$:** - $p(2)=19-36+16=-1$ (should be 5, no) 7. **Check $p(x)=29-18x+4x^2$:** - $p(2)=29-36+16=9$ (should be 5, no) **Final answer:** The polynomial $p(x)=29-18x+3x^2$ passes through all three points.