The specified function's vertex form is ![g(x) = 4(x+5)^2 - 100](https://tex.z-dn.net/?f=g%28x%29%20%3D%204%28x%2B5%29%5E2%20-%20100)
<h3>What is vertex form of a quadratic equation?</h3>
If a quadratic equation is written in the form
![y=a(x-h)^2 + k](https://tex.z-dn.net/?f=y%3Da%28x-h%29%5E2%20%2B%20k)
then it is called to be in vertex form. It is called so because when you plot this equation's graph, you will see vertex point(peak point) is on (h,k)
<h3>What is a perfect square polynomial?</h3>
If a polynomial p(x) can be written as:
![p(x) = [f(x)]^2](https://tex.z-dn.net/?f=p%28x%29%20%3D%20%5Bf%28x%29%5D%5E2)
where f(x) is also a polynomial, then p(x) is called as perfect square polynomial
For the considered case, the polynomial specified is;
![g(x) = 40x + 4x^2](https://tex.z-dn.net/?f=g%28x%29%20%20%3D%2040x%20%2B%204x%5E2)
- Case 1: Converting to vertex form
![g(x) = 40x + 4x^2\\g(x) = 4(x^2 + 10) = 4(x^2 + 10 + 25 -25)\\g(x) = 4(x^2 + 10 + 25) - 100 = 4(x+5)^2 - 100\\](https://tex.z-dn.net/?f=g%28x%29%20%20%3D%2040x%20%2B%204x%5E2%5C%5Cg%28x%29%20%3D%204%28x%5E2%20%2B%2010%29%20%3D%204%28x%5E2%20%2B%2010%20%2B%2025%20-25%29%5C%5Cg%28x%29%20%3D%204%28x%5E2%20%2B%2010%20%2B%2025%29%20-%20100%20%3D%204%28x%2B5%29%5E2%20-%20100%5C%5C)
Thus, the vertex form of the considered polynomial is ![g(x) = 4(x+5)^2 - 100](https://tex.z-dn.net/?f=g%28x%29%20%3D%204%28x%2B5%29%5E2%20-%20100)
- Case 2: Converting to standard form
Standard form of a quadratic polynomial is ![ax^2 + bx + c](https://tex.z-dn.net/?f=ax%5E2%20%2B%20bx%20%2B%20c)
Thus, we get: the considered polynomial in standard form as:
![g(x) = 4x^2 +40x](https://tex.z-dn.net/?f=g%28x%29%20%3D%204x%5E2%20%2B40x)
- Case 3: Factoring the first two terms of polynomial
![g(x) = 40x + 4x^2 \\g(x) = 4x(10 + x)](https://tex.z-dn.net/?f=g%28x%29%20%3D%2040x%20%2B%204x%5E2%20%5C%5Cg%28x%29%20%3D%204x%2810%20%2B%20x%29)
- Case 4: Forming a perfect square trinomial
The considered polynomial has only two terms, therefore, its not a trinomial.
Thus, the specified function's vertex form is ![g(x) = 4(x+5)^2 - 100](https://tex.z-dn.net/?f=g%28x%29%20%3D%204%28x%2B5%29%5E2%20-%20100)
Learn more about vertex form of a quadratic equation here:
brainly.com/question/9912128