Final answer:
To convert the given equation into standard form, we complete the square for both x and y terms and rearrange the equation, resulting in the standard form: (x + 1)² + (2y + 3)² = 16.
Step-by-step explanation:
To convert the equation x²+4y²+2x+24y+21=0 into standard form, we need to complete the square for both x and y terms.
First, we group the x terms and the y terms: (x² + 2x) + (4y² + 24y) = -21.
For the x terms, we take half of the coefficient of x, which is 1, square it, which gives us 1, and add it to both sides of the equation to complete the square: (x² + 2x + 1). We do the same process for the y terms, take half of the coefficient of y (divided by 4), which is 3, square it, which gives us 9, and add it to both sides of the equation, making sure to multiply by 4 because of the 4y² term: (4y² + 24y + 36).
Now the equation looks like this: (x² + 2x + 1) + (4y² + 24y + 36) = -21 + 1 + 36.
We simplify the constant terms on the right: -21 + 1 + 36 = 16.
The equation now reads: (x + 1)² + (2y + 3)² = 16.
Finally, we have the standard form of the equation: (x + 1)² + (2y + 3)² = 16.