We present deep photometry of the Carina dwarf spheroidal galaxy in the B and V filters from CTIO/MOSAIC out to and beyond the tidal radius of rell ≈ 0.48 degrees. The accurately calibrated photometry is combined with spectroscopic metallicity distributions of red giant branch (RGB) stars to determine the detailed star formation and chemical evolution history of Carina. The star formation history (SFH) confirms the episodic formation history of Carina and quantifies the duration and strength of each episode in great detail as a function of radius from the centre. Two main episodes of star formation occurred at old (>8 Gyr) and intermediate (2-8 Gyr) ages, both enriching stars starting from low metallicities ([Fe/H] < - 2 dex). By dividing the SFH into two components, we determine that 60 ± 9 percent of the total number of stars formed within the intermediate-age episode. Furthermore, within the tidal radius (0.48 degrees or 888 pc) a total mass in stars of 1.07 ± 0.08 × 106 M⊙ was formed, giving Carina a stellar mass-to-light ratio of 1.8 ± 0.8. By combining the detailed SFH with spectroscopic observations of RGB stars, we determined the detailed age-metallicity relation of each episode and the timescale of α-element evolution of Carina from individual stars. The oldest episode displays a tight age-metallicity relation during ≈6 Gyr with steadily declining α-element abundances and a possible α-element "knee" visible at [Fe/H] ≈ - 2.5 dex. The intermediate-age sequence displays a more complex age-metallicity relation starting from low metallicity and a sequence in α-element abundances with a slope much steeper than observed in the old episode, starting from [Fe/H] = -1.8 dex and [Mg/Fe] ≈ 0.4 dex and declining to Mg-poor values ([Mg/Fe] ≤ - 0.5 dex). This clearly indicates that the two episodes of star formation formed from gas with different abundance patterns, which is inconsistent with simple evolution in an isolated system. Tables 1-3 are only available in electronic at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/572/A10Appendices are available in electronic form at http://www.aanda.org