What is the polynomial 3*y*2 + (*y* + 7)2 – 15 after it has been fully simplified and written in standard form?\

Let's simplify the given polynomial \( 3y^2 + (y + 7)^2 - 15 \) and then write it in standard form.

### Step 1: Expand the square
First, we need to expand \( (y + 7)^2 \):
\[
(y + 7)^2 = y^2 + 2 \cdot y \cdot 7 + 7^2 = y^2 + 14y + 49
\]

### Step 2: Substitute back into the polynomial
Now substitute this back into the original polynomial:
\[
3y^2 + (y^2 + 14y + 49) - 15
\]

### Step 3: Combine like terms
Combine the \( y^2 \) terms:
\[
3y^2 + y^2 = 4y^2
\]
So, the expression becomes:
\[
4y^2 + 14y + 49 - 15
\]

Next, combine the constant terms:
\[
49 - 15 = 34
\]

### Step 4: Write the final expression
The fully simplified and standard form of the polynomial is:
\[
4y^2 + 14y + 34
\]

Would you like more details or have any questions?

Here are 5 related questions you might find helpful:
1. How do you factor a quadratic polynomial like \(4y^2 + 14y + 34\)?
2. What is the vertex form of a quadratic equation, and how is it derived?
3. Can this quadratic polynomial be solved using the quadratic formula?
4. What role does the discriminant play in determining the nature of the roots of a quadratic equation?
5. How does completing the square help in solving quadratic equations?

**Tip:** Always expand binomials carefully and combine like terms step by step to avoid mistakes in simplifying expressions.

Learn how to simplify and write the polynomial 3y^2 + (y + 7)^2 - 15 in standard form. This problem involves expanding binomials, combining like terms, and understanding polynomial operations. Suitable for high school students studying algebra and polynomial expressions.

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