Find the value of a, if (x + 2) is a factor of

Question:
Find the value of $a$, if $(x+2)$ is a factor of $4 x^{4}+2 x^{3}-3 x^{2}+8 x+5 a$

Solution:

Here, $f(x)=4 x^{4}+2 x^{3}-3 x^{2}+8 x+5 a$

By factor theorem

If (x + 2) is the factor of f(x) then, f(-2) = 0

⟹ x + 2 = 0

⟹ x = -2

Substitute the value of x in f(x)

$f(-2)=4(-2)^{4}+2(-2)^{3}-3(-2)^{2}+8(-2)+5 a$

= 4(16) + 2(-8) - 3(4) - 16 + 5a

= 64 - 16 - 12 - 16 + 5a

= 5a + 20

equate f(-2) to zero

f(-2) = 0

⟹ 5a + 20 = 0

⟹ 5a = - 20

⟹ a = -20/5

⟹ a = - 4

When a = - 4, (x + 2) will be factor of f(x)