Answer:
Factor Theorem
If (x - 4) is a factor then f(4) = 0
[tex]\begin{aligned} f(4) & = 0\\\implies(4)^2+4B-4 & =0\\12+4B & = 0\\4B & = -12\\B & = -3\end{aligned}[/tex]
[tex]\begin{aligned} f(4) & = 0\\\implies6(4)^2+4C-12 & =0\\84+4C & = 0\\4C & = -84\\C & = -21\end{aligned}[/tex]
Therefore:
[tex]\implies B+C=-3+(-21)=-24[/tex]
Part (a)
[tex]3a^2-8ab-3b^2[/tex]
[tex]\implies 3a^2+ab-9ab-3b^2[/tex]
[tex]\implies a(3a+b)-3b(3a+b)[/tex]
[tex]\implies (a-3b)(3a+b)[/tex]
Part (b)
Use the factored expression in part (a):
[tex]a=(c+d) \textsf{ and }b=(c-d)[/tex]
[tex]\implies 3(c+d)^2-8(c+d)(c-d)-3(c-d)^2[/tex]
[tex]=((c+d)-3(c-d))(3(c+d)+(c-d))[/tex]
[tex]=(c+d-3c+3d)(3c+3d+c-d)[/tex]
[tex]=(4d-2c)(4c+2d)[/tex]
