Find the value of $\frac{m(m^2+3m+3)+1}{m(m+2)+1}$ when $m=87$.

Let us simplify the numerator and the denominator expressions:Numerator:
$$m(m^2+3m+3)+1=m^3+3m^2+3m+1=(m+1{)}^3$$
Denominator:
$$m(m+2)+1=m^2+2m+1=(m+1{)}^2$$
Substituting these back into the fraction:
$$\frac{(m+1{)}^3}{(m+1{)}^2}=m+1$$
Given that $m=87$:
$$m+1=87+1=88$$