4) \( -3(4 b+1)^{\frac{3}{2}}+5=-1024 \)

Solve the equation \( -3(4b+1)^{\frac{3}{2}}+5=-1024 \). Solve the equation by following steps: - step0: Solve for \(b\): \(-3\left(4b+1\right)^{\frac{3}{2}}+5=-1024\) - step1: Find the domain: \(-3\left(4b+1\right)^{\frac{3}{2}}+5=-1024,b\geq -\frac{1}{4}\) - step2: Move the expression to the left side: \(-3\left(4b+1\right)^{\frac{3}{2}}+1029=0\) - step3: Factor the expression: \(3\left(7-4b-1\right)\left(49+7\left(4b+1\right)+\left(4b+1\right)^{2}\right)=0\) - step4: Multiply both sides: \(3\left(7-4b-1\right)\left(49+7\left(4b+1\right)+\left(4b+1\right)^{2}\right)\times \frac{1}{3}=0\times \frac{1}{3}\) - step5: Calculate: \(\left(7-4b-1\right)\left(49+7\left(4b+1\right)+\left(4b+1\right)^{2}\right)=0\) - step6: Separate into possible cases: \(\begin{align}&7-4b-1=0\\&49+7\left(4b+1\right)+\left(4b+1\right)^{2}=0\end{align}\) - step7: Solve the equation: \(\begin{align}&b=\frac{3}{2}\\&b \notin \mathbb{R}\end{align}\) - step8: Find the union: \(b=\frac{3}{2}\) - step9: Check if the solution is in the defined range: \(b=\frac{3}{2},b\geq -\frac{1}{4}\) - step10: Find the intersection: \(b=\frac{3}{2}\) - step11: Check the solution: \(b \in \varnothing \) The solution to the equation \( -3(4b+1)^{\frac{3}{2}}+5=-1024 \) is that there are no real solutions for \( b \).