= Solution
The displayed <matrix> is block diagonal with a $1\times1$ block $1$ and a $2\times2$ block $B$ having <characteristic polynomial> $X^2-4X-5=(X-5)(X+1)$. The latter block is not scalar, so its <minimal polynomial> has degree two and is exactly that <polynomial>. Taking the least common multiple of the block minimal <polynomials> gives $m=(X-1)(X-5)(X+1)$ over $\mathbb R$.
<Ideals> of $K[X]/(m)$ correspond to <ideals> of $K[X]$ containing $(m)$, namely the principal <ideals> generated by the monic divisors of $m$. Every one of the three distinct linear factors may be included or omitted, so
$$
\boxed{\#\{\text{ideals of }R\}=2^3=8.}
$$
This includes both the zero <ideal> and the whole ring. Equivalently the <Chinese remainder theorem> identifies this ring with $\mathbb R^3$.
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