Drew Brees had the most games by a quarterback with 300+ passing yards and 5+ passing touchdowns, with 9 games.

which qb has the most games with 300 passing yards and 5 tds
Interpreted as:
Which qb has the most regular-season games with 300 or more passing yards and 5 or more passing tds?
BreesBradyManningMarinoRoethlisbergerRodgersStaffordBlandaMahomesWilsonMoonKellyMontanaJohnsonJurgensenNewtonRomoMcNabbCulpepperFavreFergusonMeredithTittle98764444333333322222222
NAMEGPYDSTDCMPATTPCTAVGYDS/GTD%INTINT%SCKSCKYRATE
Drew BreesDrew Brees93,4254825532977.510.4380.614.641.2838144.6
Tom BradyTom Brady83,0074222730175.410.0375.914.010.31387144.8
Peyton ManningPeyton Manning72,7043817125766.510.5386.314.872.7534129.6
Dan MarinoDan Marino62,3333115024561.29.5388.812.752.0534123.9
Ben RoethlisbergerBen Roethlisberger41,4902210913879.010.8372.515.900.0638151.2
Aaron RodgersAaron Rodgers41,415229113070.010.9353.816.900.0426145.4
Matthew StaffordMatthew Stafford41,6142011717666.59.2403.511.463.4848119.4
George BlandaGeorge Blanda41,364227413455.210.2341.016.453.7114.6
Patrick MahomesPatrick Mahomes31,220178711675.010.5406.714.732.6434137.2
Russell WilsonRussell Wilson31,038157711368.19.2346.013.300.0960136.7
Warren MoonWarren Moon31,100157310967.010.1366.713.832.8538128.1
Jim KellyJim Kelly31,11816669966.711.3372.716.266.1546119.0
Joe MontanaJoe Montana31,333169414067.19.5444.311.432.11386126.9
Charley JohnsonCharley Johnson398617448353.011.9328.720.556.0110.2
Sonny JurgensenSonny Jurgensen31,15515539058.912.8385.016.777.8110.4
Cam NewtonCam Newton267110538661.67.8335.511.611.2440119.9
Tony RomoTony Romo281210476572.312.5406.015.411.5544147.6
Donovan McNabbDonovan McNabb280610557276.411.2403.013.900.0630152.0
Daunte CulpepperDaunte Culpepper282110628771.39.4410.511.522.3516129.5
Brett FavreBrett Favre272410527866.79.3362.012.833.8533119.9
Joe FergusonJoe Ferguson278610578567.19.2393.011.811.2130.8
Don MeredithDon Meredith275210335066.015.0376.020.000.0148.8
Y.A. TittleY.A. Tittle284613488159.310.4423.016.022.5124.3
Daniel JonesDaniel Jones13525284266.78.4352.011.900.016132.1
Deshaun WatsonDeshaun Watson14265283384.812.9426.015.200.000158.3
StatMuse has game-level data for passing touchdowns and passing yards going back to the 1950 season.

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