To convert the given quadratic expression \(y = (x + 8)(x + 11)\) into vertex form, we first need to expand it.
- Expand the expression:
\[ y = (x + 8)(x + 11) = x^2 + 11x + 8x + 88 = x^2 + 19x + 88 \]
Now, the correct expanded form of the given expression is:
\[ y = x^2 + 19x + 88 \]
- Options provided:
- \(y = x^2 + 8x + 11\)
- \(y = x^2 + 19x + 88\)
The correct option based on the expansion is:
\[ y = x^2 + 19x + 88 \]
- Convert to vertex form:
To express \(y = x^2 + 19x + 88\) in vertex form, we can complete the square.
- Start with the expression without the constant:
\[ y = x^2 + 19x + 88 \]
- To complete the square:
- Take half of the coefficient of \(x\) (which is 19), square it, and add/subtract it inside the equation.
- Half of 19 is \( \frac{19}{2} \), and squaring it gives \( \left(\frac{19}{2}\right)^2 = \frac{361}{4} \).
\[ y = \left(x^2 + 19x + \frac{361}{4}\right) - \frac{361}{4} + 88 \]
- Now combine the constants \(-\frac{361}{4} + 88 = 88 - \frac{361}{4} = \frac{352-361}{4} = \frac{-9}{4}\).
Putting it all together, we have:
\[ y = \left(x + \frac{19}{2}\right)^2 - \frac{9}{4} \]
Thus, the vertex form of the quadratic is:
\[ y = \left(x + \frac{19}{2}\right)^2 - \frac{9}{4} \]
In conclusion, among the given options, the correct expanded polynomial is:
\[ y = x^2 + 19x + 88 \]