If a 4x4 matrix A is diagonalizable, is A+4I diagonalizable as well?

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The matrix $I$ is the $4 \times 4$ identity matrix.

If this is true, how do I show that? If it's false, how would I find a counterexample?

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Hint: $$ P(D + 4I)P^{-1} = PDP^{-1} + 4I $$