Find \(f(3x+2)\)

Solution

Given

\[ f(2x+3)=4x^2+12x+15 \]

Let

\[ t=2x+3 \]

Then,

\[ x=\frac{t-3}{2} \]

Substitute into the given expression:

\[ f(t)=4\left(\frac{t-3}{2}\right)^2 +12\left(\frac{t-3}{2}\right)+15 \] \[ =(t-3)^2+6(t-3)+15 \] \[ =t^2-6t+9+6t-18+15 \] \[ =t^2+6 \]

Therefore,

\[ f(t)=t^2+6 \]

Now replace \(t\) by \(3x+2\):

\[ f(3x+2)=(3x+2)^2+6 \] \[ =9x^2+12x+4+6 \] \[ =9x^2+12x+10 \]