How do you write in standard form y=4(x+5)-2(x+1)(x+1)?

Explanation:
Since this expression has many parts, I will color-code it and tackle it one by one.
4(x+5)-2(x+1)(x+1)
For the expression in purple, all we need to do is distribute the 4 to both terms in the parenthesis. Doing this, we now have
4x+20-2(x+1)(x+1)
What I have in blue, we can multiply with the mnemonic FOIL, which stands for Firsts, Outsides, Insides, Lasts. This is the order we multiply the terms in.
Foiling (x+1)(x+1):
First terms: ${\displaystyle x\cdot x=x^2}$
Outside terms: x*1=x
Inside terms: 1*x=x
Last terms: 1*1=1
This simplifies to ${\displaystyle x^2+2x+1}$. We now have
${\displaystyle 4x+20-2(x^2+2x+1)}$
Distributing the -2 to the blue terms gives us
${\displaystyle 4x+20-2x^2-4x-2}$
Combining like terms gives us
${\displaystyle -2x^2+18}$
We see that our polynomial is in standard form, ${\displaystyle a{x}^2+bx+c}$. Notice that the x terms cancel out, so we don't have a bx term.