Certainly! Let's complete the final step of converting the given equation to standard form by completing the square.
4. Write each trinomial as a binomial squared, and simplify the right side.
We start with the equation from step 3:
[tex]\[ x^2 + 12x + 36 + y^2 + 2y + 1 = 1 + 36 + 1 \][/tex]
Now, express each trinomial as a binomial squared:
[tex]\[ x^2 + 12x + 36 = (x + 6)^2 \][/tex]
[tex]\[ y^2 + 2y + 1 = (y + 1)^2 \][/tex]
So, the left side of the equation becomes:
[tex]\[ (x + 6)^2 + (y + 1)^2 \][/tex]
The right side simplifies to:
[tex]\[ 1 + 36 + 1 = 38 \][/tex]
Therefore, the equation in standard form is:
[tex]\[ (x + 6)^2 + (y + 1)^2 = 38 \][/tex]
So, [tex]\((x + \boxed{6})^2 + (y + \boxed{1})^2 = 38\)[/tex].
