Part $1$: A continuous$-$time signal $x(t)$ has the Laplace transform

$x(s)=\frac{s+1}{s^2+5s+7}$

Determine the Laplace transform $V(s)$ of the following signals:

a$)v(t)=x(3t-4)u(3t-4)$

b$)v(t)=tx(t)$

c$)v(t)=\frac{d^{2}x(t)}{d{t}^2}$

Part $2$: By using the Laplace transform, compute the convolution $x(t)**v(t)$ where:

a$)x(t)=e^{-t}u(t),v(t)=sin(t)u(t)$

b$)x(t)=si{n}^2(t)u(t),v(t)=(t)$

Note: $si{n}^2(t)$ cam be written as:

$si{n}^2(t)=\frac{1-cos(2t)}{2}$

Part $3$: Determine the inverse Laplace transform of each of the following functions.

a$)$

$x(s)=\frac{s+2}{s^2+7s+12}$

b$)$

$x(s)=\frac{s+1}{s^3+5s^2+7s}$

c$)$

$x(s)=\frac{s+e^{-s}}{s^2+s+1}$

Write the solution $in$ a paper,please