Question

{7x^2+2x-9}/{7x+9}+(x+2)(x-3) simplify

Answer 1

To simplify the expression \(\frac{7x^2 + 2x - 9}{7x + 9} + (x + 2)(x - 3)\), we can follow these steps:

### Step 1: Simplify the second term
First, simplify \((x + 2)(x - 3)\):

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

### Step 2: Combine the terms
Now we can rewrite the expression as:

\[
\frac{7x^2 + 2x - 9}{7x + 9} + x^2 - x - 6
\]

### Step 3: Prepare to combine the fractions
To combine these, we need a common denominator, which is \(7x + 9\). Rewrite the second term with the common denominator:

\[
x^2 - x - 6 = \frac{(x^2 - x - 6)(7x + 9)}{7x + 9}
\]

### Step 4: Expand \(x^2 - x - 6\) with the denominator
Now, let's expand \((x^2 - x - 6)(7x + 9)\):

\[
= (x^2)(7x + 9) + (-x)(7x + 9) + (-6)(7x + 9)
\]
\[
= 7x^3 + 9x^2 - 7x^2 - 9x - 42x - 54
\]
\[
= 7x^3 + 2x^2 - 51x - 54
\]

### Step 5: Write combined fraction
Now, we can write our entire expression with the common denominator:

\[
\frac{7x^2 + 2x - 9 + 7x^3 + 2x^2 - 51x - 54}{7x + 9}
\]

### Step 6: Combine like terms in the numerator
Combine all of the terms in the numerator:

1. The \(x^3\) term: \(7x^3\)
2. The \(x^2\) terms: \(2x^2 + 2x^2 = 4x^2\)
3. The \(x\) terms: \(2x - 51x = -49x\)
4. The constant terms: \(-9 - 54 = -63\)

So the numerator is:

\[
7x^3 + 4x^2 - 49x - 63
\]

### Step 7: Final result
Putting the expression back together, we have:

\[
\frac{7x^3 + 4x^2 - 49x - 63}{7x + 9}
\]

This is the simplified form of the original expression. Further simplification (like factoring) would depend on the specific goals, but for now, this is an acceptable simplified form.