Solve for v. (3)/(v^2)+7v+10=(1)/(v+2)-(4)/(v+5) If there is more than one solution separate them with commas. If there is no solution, click on "No solution".

1. Simplify the Right Side<br /> Combine the fractions on the right side: $\frac{1}{v+2} - \frac{4}{v+5} = \frac{(v+5) - 4(v+2)}{(v+2)(v+5)} = \frac{v + 5 - 4v - 8}{(v+2)(v+5)} = \frac{-3v - 3}{(v+2)(v+5)}$.<br /><br />2. Equate and Simplify<br /> Set the left side equal to the simplified right side: $\frac{3}{v^2 + 7v + 10} = \frac{-3(v + 1)}{(v+2)(v+5)}$.<br /><br />3. Cross-Multiply<br /> Cross-multiply to eliminate the fractions: $3(v+2)(v+5) = -3(v+1)(v^2 + 7v + 10)$.<br /><br />4. Expand and Simplify<br /> Expand both sides: $3(v^2 + 7v + 10) = -3(v^3 + 7v^2 + 17v + 10)$.<br /><br />5. Solve for v<br /> Simplify and solve the equation: $0 = v^3 + 7v^2 + 17v + 10$. Factor or use the Rational Root Theorem to find roots.<br /><br />6. Check for Extraneous Solutions<br /> Verify solutions do not make any denominator zero.