Question

7x^2+2x-9/7x+9+(x+2)(x-3)

Answer 1

To simplify the expression \(\frac{7x^2 + 2x - 9}{7x + 9} + (x + 2)(x - 3)\), we can work through each part step by step.

First, let's simplify the quadratic expression and then evaluate the second part of the expression.

### Step 1: Factor the numerator
The expression in the numerator is \(7x^2 + 2x - 9\). We will see if it can be factored. We're looking for two numbers that multiply to \(7 \times -9 = -63\) and add up to \(2\).

The factors of \(-63\) that add up to \(2\) are \(9\) and \(-7\). Therefore, we can factor the quadratic as follows:

\[
7x^2 + 9x - 7x - 9 = (7x^2 - 7x + 9x - 9)
\]

Grouping:

\[
= 7x(x - 1) + 9(x - 1) = (7x + 9)(x - 1)
\]

Thus, the numerator can be factored as:

\[
7x^2 + 2x - 9 = (7x + 9)(x - 1)
\]

### Step 2: Substitute into the original expression
Now substitute back into the original expression:

\[
\frac{(7x + 9)(x - 1)}{7x + 9} + (x + 2)(x - 3)
\]

### Step 3: Cancel the common term
As long as \(7x + 9 \neq 0\), we can cancel this factor:

\[
= x - 1 + (x + 2)(x - 3)
\]

### Step 4: Expand the second part
Next, expand \((x + 2)(x - 3)\):

\[
(x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6
\]

### Step 5: Combine like terms
Now substitute back into the equation:

\[
= x - 1 + (x^2 - x - 6)
\]

Combine like terms:

\[
= x - 1 + x^2 - x - 6 = x^2 - 7
\]

### Conclusion
Thus, the simplified form of the original expression is:

\[
\boxed{x^2 - 7}
\]