To solve the equation [tex]\(9x^2 + 49x = 22 - 5x\)[/tex] by completing the square, we will go through the following steps:
1. First, combine like terms:
[tex]\[9x^2 + 49x = 22 - 5x\][/tex]
[tex]\[9x^2 + 54x = 22 \][/tex]
Now, divide the whole equation by 9 to simplify it:
[tex]\[x^2 + 6x = \frac{22}{9}\][/tex]
2. Next, complete the square within the parenthesis [tex]\(x^2 + 6x\)[/tex]:
To complete the square, we add and subtract [tex]\(\left(\frac{b}{2}\right)^2\)[/tex], where [tex]\(b = 6\)[/tex]:
[tex]\[\left(\frac{6}{2}\right)^2 = 9 \][/tex]
3. Add this term inside the parenthesis and to both sides:
[tex]\[x^2 + 6x + 9 - 9 = \frac{22}{9}\][/tex]
[tex]\[x^2 + 6x + 9 = \frac{22}{9} + 9 \][/tex]
[tex]\[x^2 + 6x + 9 = \frac{22}{9} + \frac{81}{9}\][/tex] (since 9 = [tex]\(\frac{81}{9}\)[/tex])
[tex]\[x^2 + 6x + 9 = \frac{103}{9} \][/tex]
By multiplying both sides by 9 to reinstate the original coefficient:
[tex]\[9(x^2 + 6x + 9) = 103 \][/tex]
Therefore, the correct step after completing the square is:
[tex]\[9\left(x^2 + 6x + 9\right) = 103 \][/tex]
Thus, the best option among the given choices is:
[tex]\[ \boxed{9\left(x^2 + 6x + 9\right) = 103} \][/tex]
