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Choice D is correct. It's given that the function $f$ is defined by $f\left(x\right)=5x^2$. Substituting $8$ for $x$ in $f\left(x\right)=5x^2$ yields $f\left(8\right)=5{\left(8\right)}^2$, which is equivalent to $f\left(8\right)=5\left(64\right)$, or $f\left(8\right)=320$. Therefore, the value of $f\left(8\right)$ is $320$.

Choice A is incorrect. This is the value of $f\left(8\right)$ if $f\left(x\right)=5x$.

Choice B is incorrect. This is the value of $f\left(8\right)$ if $f\left(x\right)=5\left(x+2\right)$.

Choice C is incorrect. This is the value of $f\left(8\right)$ if $f\left(x\right)=\left(5x\right)\left(2\right)$.