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 Problem Statement

$$ u_{tt}(x,t) + 2u_t(x,t) = u(x,t), \infty < x < \infty, t> 0$$
$$u(x,t), as x \rightarrow \infty, t>0$$
$$u(x,0) = f(x), u_t(x,0)= g(x) , \infty < x < \infty $$
 Relevant Equations

Fourier Transform equation:
$$ \frac{1}{\sqrt{2\pi}} \int_{\infty}^{\infty} f(x) e^{iwx} dx $$
I am struggling to figure out how to approach this problem. I've only solved a homogenous heat equation $$u_t = u_{xx}$$ using a fourier transform, where I can take the fourier transform of both sides then solve the general solution in fourier terms then inverse transform. However, since this question has extra terms I'm a little confused.
$$ u_{tt}(x,t) + 2u_t(x,t) = u(x,t), \infty < x < \infty, t> 0$$
$$u(x,t), as x \rightarrow \infty, t>0$$
$$u(x,0) = f(x), u_t(x,0)= g(x) , \infty < x < \infty $$
Any advice and or guidance would be greatly appreciated. All I would do is take the fourier transform of all the terms but from there I don't think I know what to do.`
$$ u_{tt}(x,t) + 2u_t(x,t) = u(x,t), \infty < x < \infty, t> 0$$
$$u(x,t), as x \rightarrow \infty, t>0$$
$$u(x,0) = f(x), u_t(x,0)= g(x) , \infty < x < \infty $$
Any advice and or guidance would be greatly appreciated. All I would do is take the fourier transform of all the terms but from there I don't think I know what to do.`