Find the value of $a$, if $x + 2$ is a factor of $4x^4 + 2x^3 - 3x^2 + 8x + 5a$.

Given:

Given expression is $4x^4 + 2x^3 - 3x^2 + 8x + 5a$.

$(x + 2)$ is a factor of $4x^4 + 2x^3 - 3x^2 + 8x + 5a$.

To do:

We have to find the value of $a$.

Solution:

We know that,

If $(x-m)$ is a root of $f(x)$ then $f(m)=0$.

$x+2=x-(-2)$

Therefore,

$f(-2)=0$

$\Rightarrow 4(-2)^4+2(-2)^3-3(-2)^2+8(-2)+5a=0$

$\Rightarrow 4(16)+2(-8)-3(4)+8(-2)+5a=0$

$\Rightarrow 64-16-12-16+5a=0$

$\Rightarrow 5a=-20$

$\Rightarrow a=\frac{-20}{5}=-4$

The value of $a$ is $-4$.