Question

\frac{7x^2+2x-9}{7x+9}+\left(x+2\right)\left(x-3\right)

Answer 1

To simplify the expression \(\frac{7x^2+2x-9}{7x+9}+\left(x+2\right)\left(x-3\right)\), we can start by simplifying each part separately.

1. First, let's simplify the product \((x+2)(x-3)\):
\[
(x+2)(x-3) = x^2 - 3x + 2x - 6 = x^2 - x - 6
\]

2. Now, we rewrite the entire expression:
\[
\frac{7x^2 + 2x - 9}{7x + 9} + x^2 - x - 6
\]

3. Next, we need a common denominator to combine the terms. The common denominator is \(7x + 9\):
\[
\frac{7x^2 + 2x - 9}{7x + 9} + \frac{(x^2 - x - 6)(7x + 9)}{7x + 9}
\]

4. We'll now expand \((x^2 - x - 6)(7x + 9)\):
\[
(x^2 - x - 6)(7x + 9) = x^2(7x + 9) - x(7x + 9) - 6(7x + 9)
\]
This becomes:
\[
7x^3 + 9x^2 - 7x^2 - 9x - 42x - 54 = 7x^3 + (9x^2 - 7x^2) - (9x + 42x) - 54
\]
Which simplifies to:
\[
7x^3 + 2x^2 - 51x - 54
\]

5. Bringing everything together, we now have:
\[
\frac{7x^2 + 2x - 9 + 7x^3 + 2x^2 - 51x - 54}{7x + 9}
\]

6. Combine all the terms in the numerator:
\[
= \frac{7x^3 + (7x^2 + 2x^2) + (2x - 51x) + (-9 - 54)}{7x + 9}
\]
Which simplifies to:
\[
= \frac{7x^3 + 9x^2 - 49x - 63}{7x + 9}
\]

At this point, you may consider factoring the numerator further or finding its roots, but this is as far as we can simplify without a specific value for \(x\).

Thus, the final expression is:
\[
\frac{7x^3 + 9x^2 - 49x - 63}{7x + 9}
\]