`(x^(2) + 2x-3) (x^(2) + 7x+5)`

Answer

To find the derivative of the function \( y = (x^2 + 2x - 3)(x^2 + 7x + 5) \), we will use the product rule of differentiation. The product rule states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative of their product is given by:

\[ \frac{d}{dx}[u \cdot v] = u \cdot \frac{dv}{dx} + v \cdot \frac{du}{dx} \]

### Step 1: Identify the functions
Let:
- \( u = x^2 + 2x - 3 \)
- \( v = x^2 + 7x + 5 \)

### Step 2: Differentiate \( u \) and \( v \)
Now we will differentiate both \( u \) and \( v \):

\[ \frac{du}{dx} = \frac{d}{dx}(x^2 + 2x - 3) = 2x + 2 \]

\[ \frac{dv}{dx} = \frac{d}{dx}(x^2 + 7x + 5) = 2x + 7 \]

### Step 3: Apply the product rule
Using the product rule:

\[ \frac{dy}{dx} = u \cdot \frac{dv}{dx} + v \cdot \frac{du}{dx} \]

Substituting \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \):

\[ \frac{dy}{dx} = (x^2 + 2x - 3)(2x + 7) + (x^2 + 7x + 5)(2x + 2) \]

### Step 4: Expand both products
Now we will expand both products:

1. Expand \( (x^2 + 2x - 3)(2x + 7) \):

\[ = x^2(2x + 7) + 2x(2x + 7) - 3(2x + 7) \]
\[ = 2x^3 + 7x^2 + 4x^2 + 14x - 6x - 21 \]
\[ = 2x^3 + 11x^2 + 8x - 21 \]

2. Expand \( (x^2 + 7x + 5)(2x + 2) \):

\[ = x^2(2x + 2) + 7x(2x + 2) + 5(2x + 2) \]
\[ = 2x^3 + 2x^2 + 14x^2 + 14x + 10x + 10 \]
\[ = 2x^3 + 16x^2 + 24x + 10 \]

### Step 5: Combine the results
Now we will combine the results from both expansions:

\[ \frac{dy}{dx} = (2x^3 + 11x^2 + 8x - 21) + (2x^3 + 16x^2 + 24x + 10) \]
\[ = 4x^3 + (11x^2 + 16x^2) + (8x + 24x) + (-21 + 10) \]
\[ = 4x^3 + 27x^2 + 32x - 11 \]

### Final Result
Thus, the derivative of the function \( y = (x^2 + 2x - 3)(x^2 + 7x + 5) \) is:

\[ \frac{dy}{dx} = 4x^3 + 27x^2 + 32x - 11 \]