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:
- The \(x^3\) term: \(7x^3\)
- The \(x^2\) terms: \(2x^2 + 2x^2 = 4x^2\)
- The \(x\) terms: \(2x - 51x = -49x\)
- 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.