The bonding in metal carbonyls involves synergic bonding, which consists of both sigma ($\sigma$) and \pi ($\pi$) character.

1. Back-Bonding Mechanism: The metal-carbon $\pi$-bond is formed by the donation of a pair of electrons from a filled $d$-orbital of the metal into the vacant antibonding $\pi ^*$ orbital of carbon monoxide. This is called back-bonding.
2. Effect on Bond Length: Back-bonding increases the electron density in the antibonding orbitals of the CO ligand. This reduces the bond order of the C—O bond, thereby increasing its bond length.
3. Role of Charge: The greater the negative charge on the central metal atom, the greater its tendency to donate electrons to the CO ligand (stronger back-bonding).

• $[Fe(CO)_4]^{2-}$: Iron has a formal charge of -2.
• $[Co(CO)_4]^-$: Cobalt has a formal charge of -1.
• $Ni(CO)_4$: Nickel has a formal charge of 0.
• $[Mn(CO)_6]^+$: Manganese has a formal charge of +1.

Since $[Fe(CO)_4]^{2-}$ has the highest negative charge on the metal, it exhibits the strongest back-bonding, the greatest reduction in C—O bond order, and consequently the longest C—O bond length .