1. **State the problem:** Simplify the expression $\frac{7x + 19}{2x - 7} + 5x + 12$. 2. **Rewrite the problem:** We want to add the rational expression $\frac{7x + 19}{2x - 7}$ to the polynomial $5x + 12$. 3. **Express the polynomial with a common denominator:** To add, write $5x + 12$ as $\frac{(5x + 12)(2x - 7)}{2x - 7}$. 4. **Multiply out the numerator:** $$ (5x + 12)(2x - 7) = 5x \cdot 2x + 5x \cdot (-7) + 12 \cdot 2x + 12 \cdot (-7) = 10x^2 - 35x + 24x - 84 = 10x^2 - 11x - 84 $$ 5. **Add the numerators:** $$ \frac{7x + 19}{2x - 7} + \frac{10x^2 - 11x - 84}{2x - 7} = \frac{7x + 19 + 10x^2 - 11x - 84}{2x - 7} = \frac{10x^2 - 4x - 65}{2x - 7} $$ 6. **Check if numerator can be factored:** Try to factor $10x^2 - 4x - 65$. 7. **Factor numerator:** Find two numbers that multiply to $10 \times (-65) = -650$ and add to $-4$. These numbers are 25 and -26 because $25 \times (-26) = -650$ and $25 + (-26) = -1$, which is not $-4$. Try other pairs. Try 10 and -65, 13 and -50, 5 and -130, none sum to -4. No integer factorization possible, so numerator is prime over integers. 8. **Final simplified expression:** $$ \frac{10x^2 - 4x - 65}{2x - 7} $$ This is the simplified form of the original expression. **Answer:** $\frac{10x^2 - 4x - 65}{2x - 7}$