Simplify. Express your answer as a single fraction in simplest form. (8)/(8b^2)+11b-5+3b+7 square

1. Factor the quadratic in the denominator <br /> $8b^{2} + 11b - 5$ <br />Multiply $8 \times (-5) = -40$. Factors of $-40$ summing to $11$ are $16$ and $-5$. <br />Rewrite: $8b^{2} + 16b - 5b - 5$ <br />Factor: $8b(b + 2) - 5(b + 2) = (8b - 5)(b + 2)$ <br /><br />2. Rewrite expression with common denominator <br /> $\frac{8}{(8b - 5)(b + 2)} + 3b + 7$ <br />Write $3b + 7$ over $(8b - 5)(b + 2)$: <br />$3b + 7 = \frac{(3b + 7)(8b - 5)(b + 2)}{(8b - 5)(b + 2)}$ <br /><br />3. Combine fractions <br /> Common denominator $(8b - 5)(b + 2)$: <br />Numerator: $8 + (3b + 7)(8b - 5)(b + 2)$ <br /><br />4. Expand numerator <br /> First $(8b - 5)(b + 2) = 8b^{2} + 16b - 5b - 10 = 8b^{2} + 11b - 10$ <br />Multiply by $(3b + 7)$: <br />$3b(8b^{2} + 11b - 10) = 24b^{3} + 33b^{2} - 30b$ <br />$7(8b^{2} + 11b - 10) = 56b^{2} + 77b - 70$ <br />Sum: $24b^{3} + (33 + 56)b^{2} + (-30 + 77)b - 70$ <br />$= 24b^{3} + 89b^{2} + 47b - 70$ <br /><br />5. Add initial numerator 8 <br /> $24b^{3} + 89b^{2} + 47b - 70 + 8 = 24b^{3} + 89b^{2} + 47b - 62$ <br /><br />6. Final fraction in simplest form <br /> $\frac{24b^{3} + 89b^{2} + 47b - 62}{(8b - 5)(b + 2)}$ — no further factoring possible.