The given function is
[tex]f(x)=2x^2+kx+19[/tex]When we divide f by (x - 4) the remainder is 35
That means If we substitute x by 4, the answer should be 35
Then we have to equate x - 4 by 0 first, to get the value of x = 4
Then substitute x by 4 and equate the answer by 35
[tex]\begin{gathered} x-4=0 \\ x-4+4=0+4 \\ x=4 \end{gathered}[/tex][tex]\begin{gathered} f(4)=35 \\ \\ 2(4)^2+k(4)+19=35 \end{gathered}[/tex]Solve the equation
[tex]\begin{gathered} 2(16)+4k+19=35 \\ \\ 32+4k+19=35 \\ \\ 51+4k=35 \end{gathered}[/tex]Subtract 51 from both sides
[tex]\begin{gathered} 51-51+4k=35-51 \\ \\ 4k=-16 \end{gathered}[/tex]Divide both sides by 4
[tex]\begin{gathered} \frac{4k}{4}=\frac{-16}{4} \\ \\ k=-4 \end{gathered}[/tex]The value of k is -4
