= Solution
Since $X$ and $P$ are <self-adjoint operators>, the <adjoint operator> relation requires $c_+=c_-=c$ and $d_+=-d_-= -d$. Thus $a_-=cX+idP$ and $a_+=cX-idP$. The <canonical commutation relation> $[X,P]=i\hbar I$ gives
$$
[a_-,a_+]=2\hbar cd\,I,\qquad
a_+a_-=c^2X^2+d^2P^2-\hbar cd\,I.
$$
Matching the $X^2$, $P^2$ and <identity operator> coefficients in $H=\hbar\omega(a_+a_-+\lambda I)$ yields
$$
c^2=\frac{m\omega}{2\hbar},\qquad
d^2=\frac1{2m\hbar\omega},\qquad cd=\frac1{2\hbar},\qquad\lambda=\frac12.
$$
Consequently all real solutions are
$$
\boxed{c_+=c_-=\varepsilon\sqrt{\frac{m\omega}{2\hbar}},\qquad
d_-=-d_+=\frac{\varepsilon}{\sqrt{2m\hbar\omega}},\qquad
\lambda=\frac12,\qquad \varepsilon\in\{1,-1\}.}
$$
The conventional <creation and annihilation operators> use $\varepsilon=1$. The other choice multiplies both by minus one and satisfies all the given equations. This harmless freedom is the <oscillator ladder-operator sign convention>; the subsequent formulas use the positive choice in both coordinates.
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