Convert the Quadratic Expression (-x-2)² - 5 to Its Standard Form

Find the standard representation of the following function

$f(x)=(-x-2)^2-5$

To find the standard representation of the quadratic function $f(x)=(-x-2)^2-5$, we'll proceed with the following steps:

• Step 1: Expand the quadratic expression $(-x-2)^2$.
• Step 2: Simplify the expression resulting from the expansion.
• Step 3: Subtract the constant term $-5$.

Let's execute these steps in detail:

Step 1: Expand the expression $(-x-2)^2$.
To expand, use the formula $(a+b)^2 = a^2 + 2ab + b^2$, where $a=-x$ and $b=-2$.
$(-x-2)^2 = (-x)^2 + 2(-x)(-2) + (-2)^2$.

Step 2: Calculate the expanded form.
$(-x)^2 = x^2$,
$2(-x)(-2) = 4x$,
and $(-2)^2 = 4$.
Combining these, we have:
$x^2 + 4x + 4$.

Step 3: Incorporate the constant from the original function.
The original function is $f(x)=(-x-2)^2-5$. Thus, we subtract 5 from the expanded result:
$f(x) = x^2 + 4x + 4 - 5$, which simplifies to:
$f(x) = x^2 + 4x - 1$.

Therefore, the standard form of the given quadratic function is $f(x) = x^2 + 4x -1$.