Answer :
ANSWER
[tex]p(x)=-3+6(x+2)^2[/tex]
EXPLANATION
The given function is
[tex]p(x)=21+24x+6x^2[/tex]
We make the coefficient of [tex]x^2[/tex] unity by factoring [tex]6[/tex] out of the last two terms to obtain;
[tex]p(x)=21+6(4x+x^2)[/tex]
We now add and subtract half the coefficient of [tex]x[/tex] multiplied by a factor of 6 to obtain;
[tex]p(x)=21-6(2)^2+6(2)^2+6(4x+x^2)[/tex]
We now factor 6 out of the last two terms to get;
[tex]p(x)=21-6\times 4+6((2)^2+4x+x^2)[/tex]
[tex]p(x)=21-24+6((2)^2+4x+x^2)[/tex]
The quadratic trinomial in the parenthesis is now a perfect square.
[tex]p(x)=-3+6(x+2)^2[/tex]
Hence the vertex form of the polynomial is
[tex]p(x)=6(x+2)^2-3[/tex]
The vertex form of p(x) will be [tex]p(x)=6(x+2)^2-3[/tex].
Given function,
[tex]p(x) = 21 + 24x + 6x^2[/tex].
We have to write p(x) in vertex form.
Now,
[tex]p(x) = 6x^2+ 24x + 21[/tex]
[tex]p(x)=6(x^{2} +4x)+21[/tex]
How to write the vertex form?
We now add and subtract half the coefficient of x multiplied by a factor of 6, we get
[tex]p(x)=6(x^{2} +4x)+2^{2} \times 6-2^{2} \times 6+21[/tex]
[tex]p(x)=6(x^{2} +4x+2^{2} )-24+21\\[/tex]
[tex]p(x)=6(x+2)^2-3[/tex]
Since the vertex form of polynomial compulsorily contains the perfect square of x.
So the vertex form of polynomial p(x) will be
[tex]p(x)=6(x+2)^2-3[/tex].
Hence the vertex form of p(x) will be [tex]p(x)=6(x+2)^2-3[/tex].
For more details about vertex form, follow the link:
https://brainly.com/question/8609063