multiply the following using the vertical multiplication method: 3x^2+5x+1* x^2-2x+4_________A. 3x^4-x^3+3x^2+x+4B. 3x^4+5x^3+10x^2+18x+4C. 3x^4-x^3+3x^2+18x+4D. 3x^4+x^3+x^2+x+4

ANSWERC. [tex]3 {x}^{4} - {x}^{3} + 3{x}^{2} + 18x + 4[/tex]EXPLANATIONWe want to multiply;[tex](3 {x}^{2} + 5x + 1)( {x}^{2} - 2x + 4)[/tex]We use the distributive property to obtain:[tex]3 {x}^{2} ( {x}^{2} - 2x + 4) + 5x( {x}^{2} - 2x + 4) + 1( {x}^{2} - 2x + 4)[/tex]We expand to get:[tex]3 {x}^{4} - 6 {x}^{3} + 12 {x}^{2} + 5 {x}^{3} - 10 {x}^{2} + 20x + {x}^{2} - 2x + 4[/tex]Regroup similar terms:[tex]3 {x}^{4} - 6 {x}^{3} + 5 {x}^{3} + 12 {x}^{2} - 10 {x}^{2} + {x}^{2} + 20x - 2x + 4[/tex]Combine similar terms:[tex]3 {x}^{4} - {x}^{3} + 3{x}^{2} + 18x + 4[/tex]The correct choice is C.