Answer:
Please check the explanation.
Step-by-step explanation:
Given the function
[tex]f\left(x\right)=\:x^2\:+8x+4[/tex]
Let's rewrite the function by completing the square.
[tex]0=x^2+8x+4[/tex]
Subtract 4 from both sides
[tex]x^2+8x+4-4=0-4[/tex]
Simplify
[tex]x^2+8x=-4[/tex]
Rewrite in the form (x+a)²=b
[tex]x^2+8x=-4[/tex]
Rewrite in the form x²+2ax+a²
solve for a, 2ax = 8x
[tex]2ax=8x[/tex]
Divide both sides by 2x
[tex]\frac{2ax}{2x}=\frac{8x}{2x}[/tex]
simplify
[tex]a = 4[/tex]
Add a²=4² to both sides
[tex]x^2+8x+4^2=-4+4^2[/tex]
[tex]x^2+8x+4^2=12[/tex]
Apply perfect square formula
(a+b)²= a²+2ab+b²
[tex]\left(x+4\right)^2=12[/tex] ∵ [tex]x^2+8x+4^2=\left(x+4\right)^2[/tex]
Thus,
[tex]\left(x+4\right)^2=12[/tex]
