In order for $f(x)$ to be a density function, we require that$$\int_{-\infty}^{+\infty}f(x)dx=1.$$

The LHS:\begin{align}\int_{-\infty}^{+\infty}f(x)dx &=\int_{-\infty}^{-1}f(x)dx+\int_{-1}^{1}f(x)dx+\int_{1}^{+\infty}f(x)dx\\
&=\int_{-\infty}^{-1}0\ dx+\int_{-1}^{1}c(1-x^2)dx+\int_{1}^{+\infty}0\ dx\\
&=c\int_{-1}^{1}(1-x^2)dx\\
&=c\left(\int_{-1}^{1}dx-\int_{-1}^{1}x^2dx\right)\\
&=c\left(x\Bigg\rvert_{-1}^1-\frac{x^3}{3}\Bigg\rvert_{-1}^1\right)\\
&=c\left(1-(-1)-\frac{1}{3}+\frac{-1}{3}\right)\\
&=c\left(2-\frac{2}{3}\right)\\
&=c\frac{4}{3}.\end{align}

Making LHS=RHS, we conclude that $c=\frac{3}{4}$.