ANSWER
The remainder is 15.
EXPLANATION
The given polynomial function is
[tex]p(x) = 3 {x}^{4} + 2 {x}^{3} - {x}^{2} + 2x - 9[/tex]
We want to find the remainder when the given polynomial function is divided by
[tex]x + 2[/tex]
According to the Remainder Theorem, if
[tex]p(x) = 3 {x}^{4} + 2 {x}^{3} - {x}^{2} + 2x - 9[/tex]
is divided by
[tex]x + 2[/tex]
Then the remainder is
[tex]p( - 2)[/tex]
This means that, we have to substitute
[tex]x = - 2[/tex]
into the given polynomial function and evaluate.
Thus,
[tex]p( - 2) = 3 {( - 2)}^{4} + 2 {( - 2)}^{3} - {( - 2)}^{2} + 2( - 2)- 9[/tex]
[tex]p( - 2) = 3 {( 16)} + 2 {( - 8)} - {( 4)} + 2( - 2)- 9[/tex]
[tex]p( - 2) = 48 - 16- 4 - 4- 9[/tex]
[tex]p( - 2) = 15[/tex]
Therefore the remainder is 15
