1. **Stating the problem:** We have a rectangular prism with side lengths $x$, $x+2$, and $x+3$. The volume is given as 90 cm³. 2. **Formula used:** The volume $V$ of a rectangular prism is calculated by multiplying its length, width, and height: $$V = x \times (x+2) \times (x+3)$$ 3. **Set up the equation:** Given $V = 90$, we have: $$x(x+2)(x+3) = 90$$ 4. **Expand the expression:** First, expand $(x+2)(x+3)$: $$ (x+2)(x+3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6 $$ So the equation becomes: $$ x(x^2 + 5x + 6) = 90 $$ 5. **Simplify:** $$ x^3 + 5x^2 + 6x = 90 $$ 6. **Bring all terms to one side:** $$ x^3 + 5x^2 + 6x - 90 = 0 $$ 7. **Solve for $x$:** We look for integer roots by testing factors of 90. Try $x=3$: $$3^3 + 5(3)^2 + 6(3) - 90 = 27 + 45 + 18 - 90 = 0$$ Since this equals zero, $x=3$ is a root. 8. **Answer:** The value of $x$ is 3. **Final answer:** $\boxed{3}$