Find the derivative of the function $f(x)=2 x^{2}+3 x-5$ at $x=-1$. Also prove that $f^{\prime}(0)+3 f^{\prime}(-1)=0$.

Step 1: Find the general derivative of $f(x)$

To find the derivative of the function $f(x) = 2x^2 + 3x - 5$, we apply the power rule and the constant multiple rule of differentiation. The derivative of $2x^2$ is $4x$, the derivative of $3x$ is $3$, and the derivative of a constant $-5$ is $0$.

Step 2: Evaluate $f'(-1)$

Now we substitute $x = -1$ into the derivative function $f'(x) = 4x + 3$ to find the derivative at that specific point.

Step 3: Evaluate $f'(0)$

Similarly, we substitute $x = 0$ into the derivative function $f'(x) = 4x + 3$ to find the derivative at $x=0$.

Step 4: Prove the given relation

Finally, we substitute the calculated values of $f'(0)$ and $f'(-1)$ into the given relation $f'(0) + 3f'(-1)$ to verify if it equals zero.

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